The Database of Integer Sequences, Part 130
Part of the On-Line Encyclopedia of Integer Sequences
This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.
For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )
Seis.html: Welcome
index.html: Lookup
indexfr.html: Francais
demo1.html: Demos
Sindx.html: Index
WebCam.html: WebCam
Submit.html: Contribute new sequence or comment
eishelp1.html: Internal format
eishelp2.html: Beautified format
transforms.html: Transforms
Spuzzle.html: Puzzles
Shot.html: Hot
classic.html: Classics
ol.html: Superseeker
JIS/index.html: Journal of Integer Sequences
pages.html: More pages
Maintained by: N. J. A. Sloane (njas@research.att.com),
home page: www.research.att.com/~njas/
(start)
%I A142409
%S A142409 83,131,179,227,419,467,563,659,947,1091,1187,1283,1427,1523,1571,1619,
%T A142409 1667,1811,1907,2003,2099,2243,2339,2531,2579,2819,2963,3011,3203,3251,
%U A142409 3299,3347,3491,3539,3779,3923,4019,4211,4259,4451,4547,4643,4691,4787
%N A142409 Primes congruent to 35 mod 48.
%Y A142409 Sequence in context: A033252 A096279 A142309 this_sequence A140038 A126711 A039548
%Y A142409 Adjacent sequences: A142406 A142407 A142408 this_sequence A142410 A142411 A142412
%K A142409 nonn
%O A142409 1,1
%A A142409 njas, Jul 11 2008
%I A140038
%S A140038 83,131,227,563,1091,1427,1811,1931,1979,2243,2411,2939,3251,3659,3779,
%T A140038 3923,4091,4259,4451,4787,5099,5507,5843,5939,6299,6947,6971,7523,7691,
%U A140038 8147,8291,8819,9203,9323,9371,9467,9539,9803,10139,10163
%N A140038 Primes of the form 24x^2+24xy+83y^2.
%C A140038 Discriminant=-7392. See A139827 for more information.
%F A140038 The primes are congruent to {83, 131, 227, 299, 395, 563, 635, 755, 899, 923, 1091, 1139, 1403, 1427, 1811} (mod 1848).
%Y A140038 Sequence in context: A096279 A142309 A142409 this_sequence A126711 A039548 A141976
%Y A140038 Adjacent sequences: A140035 A140036 A140037 this_sequence A140039 A140040 A140041
%K A140038 nonn,easy
%O A140038 1,1
%A A140038 T. D. Noe (noe(AT)sspectra.com), May 02 2008
%I A126711
%S A126711 83,137,191,227,299,317,353,443,461,587,821,827,839,857,877,977,1031,
%T A126711 1091,1109,1163,1277,1289,1307,1367,1427,1433,1451,1523,1619,1627,1667,
%U A126711 1787,1811,1847,1913,1973,1997,2243,2333,2377,2417,2543,2621,2657,2693
%N A126711 Primes of the form pqrs+2 with p,q,r,s odd primes.
%F A126711 {A014613(i)+2 such that A014613(i)+2 is in A000040}.
%e A126711 a(1) = 83 = 3*3*3*3+2.
%e A126711 a(2) = 137 = 3*3*3*5+2.
%e A126711 a(3) = 191 = 3*3*3*7+2.
%e A126711 a(4) = 227 = 3*3*5*5+2.
%Y A126711 Cf. A000040, A014613, A126608-A126609, A126636, A126660-A126661.
%Y A126711 Sequence in context: A142309 A142409 A140038 this_sequence A039548 A141976 A142652
%Y A126711 Adjacent sequences: A126708 A126709 A126710 this_sequence A126712 A126713 A126714
%K A126711 easy,nonn
%O A126711 1,1
%A A126711 Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 12 2007
%I A039548
%S A039548 83,138,227,282,371,426,515,570,659,714,803,858,875,887,899,911,923,
%T A039548 935,959,971,983,995,996,997,998,999,1000,1001,1003,1004,1005,1006,
%U A039548 1091,1146,1235,1290,1379,1434,1523,1578,1590,1602,1614,1626,1638
%N A039548 Numbers n such that representation in base 12 has same nonzero number of 6's and 11's.
%Y A039548 Sequence in context: A142409 A140038 A126711 this_sequence A141976 A142652 A140543
%Y A039548 Adjacent sequences: A039545 A039546 A039547 this_sequence A039549 A039550 A039551
%K A039548 nonn,base,easy
%O A039548 1,1
%A A039548 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A141976
%S A141976 83,139,167,223,251,307,419,503,587,643,727,811,839,1063,1091,1231,1259,
%T A141976 1399,1427,1483,1511,1567,1847,1931,1987,2099,2239,2267,2351,2659,2687,
%U A141976 2939,3023,3079,3163,3191,3331,3359,3499,3527,3583,3779,3863,3919,3947
%N A141976 Primes congruent to 27 mod 28.
%Y A141976 Sequence in context: A140038 A126711 A039548 this_sequence A142652 A140543 A141867
%Y A141976 Adjacent sequences: A141973 A141974 A141975 this_sequence A141977 A141978 A141979
%K A141976 nonn
%O A141976 1,1
%A A141976 njas, Jul 11 2008
%I A142652
%S A142652 83,139,251,307,419,587,643,811,1091,1259,1427,1483,1931,1987,2099,2267,
%T A142652 2659,2939,3163,3331,3499,3779,3947,4003,4283,4339,4451,4507,4787,5011,
%U A142652 5179,5347,5683,5851,6131,6299,6691,6803,6971,7027,7307,7643,7699,7867
%N A142652 Primes congruent to 27 mod 56.
%Y A142652 Sequence in context: A126711 A039548 A141976 this_sequence A140543 A141867 A142118
%Y A142652 Adjacent sequences: A142649 A142650 A142651 this_sequence A142653 A142654 A142655
%K A142652 nonn
%O A142652 1,1
%A A142652 njas, Jul 11 2008
%I A142118
%S A142118 83,157,379,601,823,971,1193,1489,1637,1933,2081,2377,2969,3191,3413,3709,
%T A142118 3931,4079,4153,4523,4597,4967,5189,6151,6299,6373,6521,7039,7187,7853,
%U A142118 7927,8297,8741,8963,9629,9851,10369,10739,11257,11701,11923,12071,12589
%N A142118 Primes congruent to 9 mod 37.
%Y A142118 Sequence in context: A142652 A140543 A141867 this_sequence A044253 A044634 A136079
%Y A142118 Adjacent sequences: A142115 A142116 A142117 this_sequence A142119 A142120 A142121
%K A142118 nonn
%O A142118 1,1
%A A142118 njas, Jul 11 2008
%I A044253
%S A044253 83,164,245,326,407,488,569,650,731,747,812,893,974,1055,1136,
%T A044253 1217,1298,1379,1460,1476,1541,1622,1703,1784,1865,1946,2027,
%U A044253 2108,2189,2205,2270,2351,2432,2513,2594,2675,2756,2837,2918
%N A044253 Numbers n such that string 0,2 occurs in the base 9 representation of n but not of n-1.
%Y A044253 Sequence in context: A140543 A141867 A142118 this_sequence A044634 A136079 A118359
%Y A044253 Adjacent sequences: A044250 A044251 A044252 this_sequence A044254 A044255 A044256
%K A044253 nonn,base
%O A044253 1,1
%A A044253 Clark Kimberling (ck6(AT)evansville.edu)
%I A044634
%S A044634 83,164,245,326,407,488,569,650,731,755,812,893,974,1055,1136,1217,
%T A044634 1298,1379,1460,1484,1541,1622,1703,1784,1865,1946,2027,2108,2189,2213,
%U A044634 2270,2351,2432,2513,2594,2675,2756,2837,2918
%N A044634 Numbers n such that string 0,2 occurs in the base 9 representation of n but not of n+1.
%Y A044634 Sequence in context: A141867 A142118 A044253 this_sequence A136079 A118359 A084866
%Y A044634 Adjacent sequences: A044631 A044632 A044633 this_sequence A044635 A044636 A044637
%K A044634 nonn,base
%O A044634 1,1
%A A044634 Clark Kimberling (ck6(AT)evansville.edu)
%I A136079
%S A136079 83,167,251,293,419,503,797,881,1259,1301,1427,1511,1553,1889,2141,2267,
%T A136079 2309,2393,2687,2897,2939,3191,3527,3779,3821,4073,4157,4451,4703,4787,
%U A136079 5039,5081,5417,5669,5711,6173,6551,6971,7307,7349,7433,7559,7727,7853
%N A136079 Father primes of order 10.
%C A136079 For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075. For father primes of order 7 see A136076. For father primes of order 8 see A136077. For father primes of order 9 see A136078
%t A136079 n = 10; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a
%Y A136079 Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, A136080.
%Y A136079 Sequence in context: A142118 A044253 A044634 this_sequence A118359 A084866 A142332
%Y A136079 Adjacent sequences: A136076 A136077 A136078 this_sequence A136080 A136081 A136082
%K A136079 nonn
%O A136079 1,1
%A A136079 Artur Jasinski (grafix(AT)csl.pl), Dec 12 2007
%I A118359
%S A118359 83,167,251,433,503,587,601,727,1063,1217,1231,1553,1777,1861,1973,1987,
%T A118359 2281,2351,2393,2897,3541,4073,4283,4451,4507,4591,4871,5081,5431,5557,
%U A118359 5641,5683
%N A118359 Primes for which the weight as defined in A117078 is 7 and the gap as defined in A001223 is 6.
%C A118359 The prime numbers in this sequence are of the form (14i-1) with i=(level(n)+1)/2, level(n) defined in A117563. level(n) is not multiple of 3.
%H A118359 Remi Eismann, Table of n, a(n) for n = 1..10000
%e A118359 prime(24) = prime (23) + prime(23)mod(7) = prime (23) + prime(23)mod(77)
%e A118359 89 = 83 + 83mod(7) = 83 + 83mod(77)
%e A118359 k=7, level = 77/7 = 11
%Y A118359 Cf. A117078, A117563, A001359, A074822, A118922, A118924, A119504, A119597, A119596, A119595.
%Y A118359 Sequence in context: A044253 A044634 A136079 this_sequence A084866 A142332 A111078
%Y A118359 Adjacent sequences: A118356 A118357 A118358 this_sequence A118360 A118361 A118362
%K A118359 nonn
%O A118359 1,1
%A A118359 Remi EISMANN (reismann(AT)free.fr), May 24 2006, May 04 2007
%I A084866
%S A084866 83,173,197,269,317,389,461,557,653,701,797,941,1091,1109,1181,1229,
%T A084866 1637,1709,1949,1997,2069,2141,2309,2531,2549,2621,2789,2861,3221,3389,
%U A084866 3461,3581,3821,4157,4229,4349,4493,5051,5261,5381,5501,5693
%N A084866 Primes that can be written in the form 2*p^2 + 3*q^2 with p and q prime.
%C A084866 Subsequence of A084864 and of A084865; A084863(a(n))>0.
%e A084866 A000040(40) = 173 = 98 + 75 = 2*7^2 + 3*5^2 = 2*A000040(4)^2 +
%e A084866 3*A000040(3)^2, therefore 173 is a term.
%Y A084866 Sequence in context: A044634 A136079 A118359 this_sequence A142332 A111078 A106962
%Y A084866 Adjacent sequences: A084863 A084864 A084865 this_sequence A084867 A084868 A084869
%K A084866 nonn
%O A084866 1,1
%A A084866 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 10 2003
%I A142332
%S A142332 83,173,263,353,443,983,1163,1433,1523,1613,1973,2063,2153,2243,2333,2423,
%T A142332 2693,2963,3323,3413,3593,3863,4133,4493,4583,4673,4943,5303,5393,5483,
%U A142332 5573,5843,6113,6203,6473,6563,6653,6833,7013,7103,7193,7283,7643,7823
%N A142332 Primes congruent to 38 mod 45.
%Y A142332 Sequence in context: A136079 A118359 A084866 this_sequence A111078 A106962 A137364
%Y A142332 Adjacent sequences: A142329 A142330 A142331 this_sequence A142333 A142334 A142335
%K A142332 nonn
%O A142332 1,1
%A A142332 njas, Jul 11 2008
%I A111078
%S A111078 0,83,174,276,388,512,650,801,969,1154,1358,1584,1833,2107,2411,2746,
%T A111078 3115,3523,3973,4470,5018,5624,6291,7028,7842,8740,9730,10824,12031,
%U A111078 13363,14833,16456,18247,20224,22406,24815,27473,30408,33648,37224
%N A111078 Concerning the popular MMORPG "Runescape" by JAGeX corporation, this sequence gives the amount of experience points needed for a given level in a skill.
%H A111078 JAGeX, Runescape.
%H A111078 Author?, XP table and formula
%F A111078 xp(n) = floor( sum( floor(x + 300*2^(x/7), x, 1, n-1) ) / 4 ) Notation: sum(equation, variable, first index, last index)
%Y A111078 Sequence in context: A118359 A084866 A142332 this_sequence A106962 A137364 A106094
%Y A111078 Adjacent sequences: A111075 A111076 A111077 this_sequence A111079 A111080 A111081
%K A111078 dumb,nonn
%O A111078 1,2
%A A111078 Daniel Hayes (swedishlf(AT)hotmail.com), Oct 11 2005
%I A106962
%S A106962 83,179,223,397,479,541,563,677,727,757,811,863,907,941,967,1103,1277,
%T A106962 1289,1433,1489,1523,1553,1867,1889,1993,1997,2011,2039,2311,2341,2383,
%U A106962 2459,2551,2579,2633,2647,2917,2999,3271,3373,3469,3739,3797
%N A106962 Primes of the form x^2+xy+20y^2, with x and y nonnegative.
%C A106962 Discriminant=-79. See A106856 for more information.
%Y A106962 Sequence in context: A084866 A142332 A111078 this_sequence A137364 A106094 A142443
%Y A106962 Adjacent sequences: A106959 A106960 A106961 this_sequence A106963 A106964 A106965
%K A106962 nonn,easy
%O A106962 1,1
%A A106962 T. D. Noe (noe(AT)sspectra.com), May 09 2005
%I A137364
%S A137364 83,179,227,347,419,419,467,491,563,587,659,659,827,971,1019,1019,1091,
%T A137364 1259,1427,1499,1499,1667,1811,1811,1907,1907,1979,1979,2027,2243,2267,
%U A137364 2339,2339,2531,2579,2699,2819,2843,2939,3347,3539,3539,3659,3659,3779
%N A137364 Prime numbers n such that n = p1^2 + p2^2 + p3^2, a sum of squares of 3 distinct prime numbers.
%C A137364 Multiple solutions with different sets {p1,p2,p3} are indicated by repeating the entry for each solution. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 12 2008
%e A137364 83=3^2+5^2+7^2;
%e A137364 179=3^2+7^2+11^2;
%e A137364 227=3^2+7^2+13^2.
%t A137364 Array[r, 99]; Array[y, 99]; For[i = 0, i < 10^2, r[i] = y[i] = 0; i++ ]; z = 4^2; n = 0; For[i1 = 1, i1 < z, a = Prime[i1]; a2 = a^2; For[i2 = i1 + 1, i2 < z, b = Prime[i2]; b2 = b^2; For[i3 = i2 + 1, i3 < z, c = Prime[i3]; c2 = c^2; p = a2 + b2 + c2; If[PrimeQ[p], Print[a2, " + ", b2, " + ", c2, " = ", p]; n++; r[n] = p]; i3++ ]; i2++ ]; i1++ ]; Sort[Array[r, 39]]
%Y A137364 Sequence in context: A142332 A111078 A106962 this_sequence A106094 A142443 A044415
%Y A137364 Adjacent sequences: A137361 A137362 A137363 this_sequence A137365 A137366 A137367
%K A137364 nonn
%O A137364 1,1
%A A137364 Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 09 2008
%E A137364 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 12 2008
%I A106094
%S A106094 83,181,281,283,383,487,587,683,787,811,821,823,827,853,857,863,877,881,
%T A106094 883,887,1087,1181,1187,1283,1381,1481,1483,1487,1583,1783,1787,1801,
%U A106094 1811,1823,1831,1847,1861,1867,1871,1873,1877
%N A106094 Primes with maximal digit = 8.
%t A106094 Select[Prime[Range[200]], Max[IntegerDigits[ # ]]==8&]
%Y A106094 Sequence in context: A111078 A106962 A137364 this_sequence A142443 A044415 A044796
%Y A106094 Adjacent sequences: A106091 A106092 A106093 this_sequence A106095 A106096 A106097
%K A106094 nonn,base
%O A106094 1,1
%A A106094 Zak Seidov (zakseidov(AT)yahoo.com), May 07 2005
%I A142443
%S A142443 83,181,769,1063,1259,1553,1847,2141,2239,2729,3023,3121,3709,4003,4297,
%T A142443 4493,4591,4787,5081,5179,5669,6257,6551,7237,7433,7727,8609,8707,9001,
%U A142443 9491,9883,10079,10177,10667,11059,11353,11549,11941,12823,13313,13411
%N A142443 Primes congruent to 34 mod 49.
%Y A142443 Sequence in context: A106962 A137364 A106094 this_sequence A044415 A044796 A106900
%Y A142443 Adjacent sequences: A142440 A142441 A142442 this_sequence A142444 A142445 A142446
%K A142443 nonn
%O A142443 1,1
%A A142443 njas, Jul 11 2008
%I A044415
%S A044415 83,183,283,383,483,583,683,783,830,883,983,1083,1183,1283,1383,
%T A044415 1483,1583,1683,1783,1830,1883,1983,2083,2183,2283,2383,2483,
%U A044415 2583,2683,2783,2830,2883,2983,3083,3183,3283,3383,3483,3583
%N A044415 Numbers n such that string 8,3 occurs in the base 10 representation of n but not of n-1.
%Y A044415 Sequence in context: A137364 A106094 A142443 this_sequence A044796 A106900 A142621
%Y A044415 Adjacent sequences: A044412 A044413 A044414 this_sequence A044416 A044417 A044418
%K A044415 nonn,base
%O A044415 1,1
%A A044415 Clark Kimberling (ck6(AT)evansville.edu)
%I A044796
%S A044796 83,183,283,383,483,583,683,783,839,883,983,1083,1183,1283,1383,1483,
%T A044796 1583,1683,1783,1839,1883,1983,2083,2183,2283,2383,2483,2583,2683,2783,
%U A044796 2839,2883,2983,3083,3183,3283,3383,3483,3583
%N A044796 Numbers n such that string 8,3 occurs in the base 10 representation of n but not of n+1.
%Y A044796 Sequence in context: A106094 A142443 A044415 this_sequence A106900 A142621 A142681
%Y A044796 Adjacent sequences: A044793 A044794 A044795 this_sequence A044797 A044798 A044799
%K A044796 nonn,base
%O A044796 1,1
%A A044796 Clark Kimberling (ck6(AT)evansville.edu)
%I A106900
%S A106900 83,191,197,269,439,487,523,619,823,907,947,977,1193,1277,1319,1447,
%T A106900 1481,1499,1579,1709,1741,1811,1861,1867,2053,2213,2221,2273,2351,2447,
%U A106900 2539,2777,2789,2879,2917,2939,2963,3061,3089,3203,3373,3449
%N A106900 Primes of the form x^2+xy+12y^2, with x and y nonnegative.
%C A106900 Discriminant=-47. See A106856 for more information.
%Y A106900 Sequence in context: A142443 A044415 A044796 this_sequence A142621 A142681 A066366
%Y A106900 Adjacent sequences: A106897 A106898 A106899 this_sequence A106901 A106902 A106903
%K A106900 nonn,easy
%O A106900 1,1
%A A106900 T. D. Noe (noe(AT)sspectra.com), May 09 2005
%I A142621
%S A142621 83,193,523,743,853,1733,2063,2393,2503,2833,3163,3823,4153,4373,4483,4703,
%T A142621 4813,5693,6133,6353,6793,7013,7673,8443,8663,9103,9323,9433,10093,10313,
%U A142621 10753,10973,11083,11633,11743,12073,12953,13063,13613,13723,14713,15263
%N A142621 Primes congruent to 28 mod 55.
%Y A142621 Sequence in context: A044415 A044796 A106900 this_sequence A142681 A066366 A142001
%Y A142621 Adjacent sequences: A142618 A142619 A142620 this_sequence A142622 A142623 A142624
%K A142621 nonn
%O A142621 1,1
%A A142621 njas, Jul 11 2008
%I A142681
%S A142681 83,197,311,653,881,1109,1223,1451,1907,2477,2591,2819,3389,3617,4073,4643,
%T A142681 4871,5099,5441,5669,5783,5897,6011,6353,6581,7151,7607,7949,8291,8747,
%U A142681 8861,9203,9431,9887,10343,10457,10799,11027,11369,11483,11597,11939,12281
%N A142681 Primes congruent to 26 mod 57.
%Y A142681 Sequence in context: A044796 A106900 A142621 this_sequence A066366 A142001 A065012
%Y A142681 Adjacent sequences: A142678 A142679 A142680 this_sequence A142682 A142683 A142684
%K A142681 nonn
%O A142681 1,1
%A A142681 njas, Jul 11 2008
%I A066366
%S A066366 83,199,223,251,857,863,883,941,983,991,1061,1151,1187,1283,1367,1381,
%T A066366 1433,1439,1523,1553,1607,1753,1901,2011,2179,2357,2393,2647,2689,2731,
%U A066366 2777,2837,2927,2963,3037,3121,3181,3617,3821,3853,3911,3967,4217,4337
%N A066366 Primes which are the sum of a prime number of consecutive primes in a prime number of different ways.
%H A066366 G. L. Honaker, Jr. and C. K. Caldwell, Prime Curios! 83
%e A066366 199 is on the list because it can be written in two (prime) ways as the sum of either three (prime) or five (prime) consecutive primes: 61+67+71 = 31+37+41+43+47.
%Y A066366 Cf. A065867.
%Y A066366 Sequence in context: A106900 A142621 A142681 this_sequence A142001 A065012 A033253
%Y A066366 Adjacent sequences: A066363 A066364 A066365 this_sequence A066367 A066368 A066369
%K A066366 nonn
%O A066366 1,1
%A A066366 Henry Bottomley (se16(AT)btinternet.com), Dec 21 2001
%I A142001
%S A142001 83,199,257,373,431,547,953,1069,1301,1823,1997,2113,2287,2693,3041,3331,
%T A142001 3389,3853,3911,4027,4201,4259,4549,4723,5303,5419,5477,5651,6173,6521,
%U A142001 6637,6869,7043,7159,7333,7507,7681,8087,8377,8609,8783,9421,9479,9769
%N A142001 Primes congruent to 25 mod 29.
%Y A142001 Sequence in context: A142621 A142681 A066366 this_sequence A065012 A033253 A141933
%Y A142001 Adjacent sequences: A141998 A141999 A142000 this_sequence A142002 A142003 A142004
%K A142001 nonn
%O A142001 1,1
%A A142001 njas, Jul 11 2008
%I A065012
%S A065012 83,199,328,365,534,581,735,790,847,1031,1096,1163,1304,1377,1452,1529,
%T A065012 1690,1773,1858,1945,2035,2126,2219,2314,2411,2511,2612,2715,2820,2927,
%U A065012 2928,3037,3148,3261,3376,3493,3494,3613,3734,3857,3982,3983,4109,4110
%N A065012 Integers for which the periodic part of the continued fraction for the square root of n begins with 9.
%e A065012 The continued fraction for the square root of 83 is 9, {9, 18}.
%t A065012 Select[ Range[5000], First[ Last[ ContinuedFraction[ Sqrt[ # ]]]] == 9 & ]
%Y A065012 Sequence in context: A142681 A066366 A142001 this_sequence A033253 A141933 A008897
%Y A065012 Adjacent sequences: A065009 A065010 A065011 this_sequence A065013 A065014 A065015
%K A065012 cofr,easy,nonn
%O A065012 1,1
%A A065012 Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 01 2001
%I A033253
%S A033253 83,227,557,659,751,773,811,983,1061,1231,1409,1423,1531,
%T A033253 1553,1847,2111,2347,2357,2399,2417,2683,2999,3037,3109,
%U A033253 3371,3517,3581,3613,3929,4111,4211,4357,4391,4591,4643
%N A033253 Primes of form x^2+83*y^2.
%D A033253 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%Y A033253 Sequence in context: A066366 A142001 A065012 this_sequence A141933 A008897 A023284
%Y A033253 Adjacent sequences: A033250 A033251 A033252 this_sequence A033254 A033255 A033256
%K A033253 nonn
%O A033253 1,1
%A A033253 njas
%I A141933
%S A141933 83,233,283,383,433,683,733,883,983,1033,1283,1433,1483,1583,1733,1783,
%T A141933 1933,2083,2333,2383,2633,2683,2833,3083,3433,3533,3583,3733,3833,4133,
%U A141933 4283,4483,4583,4733,4783,4933,5233,5333,5483,5683,5783,6133,6733,6833
%N A141933 Primes congruent to 8 mod 25.
%Y A141933 Sequence in context: A142001 A065012 A033253 this_sequence A008897 A023284 A142025
%Y A141933 Adjacent sequences: A141930 A141931 A141932 this_sequence A141934 A141935 A141936
%K A141933 nonn
%O A141933 1,1
%A A141933 njas, Jul 11 2008
%I A008897
%S A008897 83,248,124,62,31,92,46,23,68,34,17,50,25,74,37,110,55,
%T A008897 164,82,41,122,61,182,91,272,136,68,34,17,50,25,74,37,110,
%U A008897 55,164,82,41,122,61,182,91,272,136,68,34,17,50,25,74,37
%N A008897 x->x/2 if x even, x->3x-1 if x odd.
%D A008897 R. K. Guy, Unsolved Problems in Number Theory, E16.
%Y A008897 Sequence in context: A065012 A033253 A141933 this_sequence A023284 A142025 A142387
%Y A008897 Adjacent sequences: A008894 A008895 A008896 this_sequence A008898 A008899 A008900
%K A008897 nonn
%O A008897 0,1
%A A008897 njas
%I A023284
%S A023284 83,263,1217,1319,1511,1721,1847,1907,2141,2531,4283,4673,5333,6089,6353,
%T A023284 6983,7013,7151,7529,8543,10709,10973,11423,15077,15137,17387,17573,
%U A023284 20129,20201,20411,20663,21521,23369,23561,25343,26669,27143,27647,28697
%N A023284 Numbers n such that n remains prime through 3 iterations of function f(x) = 5x + 4.
%Y A023284 Sequence in context: A033253 A141933 A008897 this_sequence A142025 A142387 A031433
%Y A023284 Adjacent sequences: A023281 A023282 A023283 this_sequence A023285 A023286 A023287
%K A023284 nonn
%O A023284 1,1
%A A023284 David W. Wilson (davidwwilson(AT)comcast.net)
%I A142025
%S A142025 83,269,331,641,827,1013,1447,1571,2129,2377,2687,2749,3121,3307,3617,3803,
%T A142025 3989,4051,4423,4547,4733,4919,5167,5477,5849,6221,6469,6779,6841,7027,
%U A142025 7151,7213,7523,8081,8329,8887,9011,9631,9817,9941,10313,10499,11057,11119
%N A142025 Primes congruent to 21 mod 31.
%Y A142025 Sequence in context: A141933 A008897 A023284 this_sequence A142387 A031433 A061525
%Y A142025 Adjacent sequences: A142022 A142023 A142024 this_sequence A142026 A142027 A142028
%K A142025 nonn
%O A142025 1,1
%A A142025 njas, Jul 11 2008
%I A142387
%S A142387 83,271,647,929,1117,1399,1493,2339,2621,2903,3373,3467,4219,4783,4877,
%T A142387 5347,5441,6287,6569,7039,7321,7603,8167,8543,8731,9013,9859,10141,10799,
%U A142387 10987,11551,11833,11927,12491,13337,13619,13807,13901,14653,14747,15217
%N A142387 Primes congruent to 36 mod 47.
%Y A142387 Sequence in context: A008897 A023284 A142025 this_sequence A031433 A061525 A142496
%Y A142387 Adjacent sequences: A142384 A142385 A142386 this_sequence A142388 A142389 A142390
%K A142387 nonn
%O A142387 1,1
%A A142387 njas, Jul 11 2008
%I A031433
%S A031433 83,328,735,1304,2035,2928,3983,5200,6579,7415,8120,9235,9823,11688,
%T A031433 13715,15904,17190,18255,20768,23443,23750,26280,26605,29279,29622,
%U A031433 31015,32440,35763,39248,42895,45416,46704,48890,50675,54808,59103
%N A031433 Least term in period of continued fraction for sqrt(n) is 9.
%Y A031433 Sequence in context: A023284 A142025 A142387 this_sequence A061525 A142496 A142560
%Y A031433 Adjacent sequences: A031430 A031431 A031432 this_sequence A031434 A031435 A031436
%K A031433 nonn
%O A031433 1,1
%A A031433 David W. Wilson (davidwwilson(AT)comcast.net)
%I A061525
%S A061525 83,361,951,1997,3667,6153,9671,14461,20787,28937,39223,51981,67571,
%T A061525 86377,108807,135293,166291,202281,243767,291277,345363,406601,475591,
%U A061525 552957,639347,735433,841911,959501,1088947,1231017,1386503,1556221
%N A061525 Surround numbers of an n X 2 rectangle when n is odd.
%C A061525 See A061524 for even n.
%H A061525 E. J. Friedman, Math. Magic: May, 1999
%F A061525 a(n) = (n^4 + 32n^3 + 278n^2 + 656n + 361)/16 when n is odd.
%Y A061525 Cf. A047875 A061524.
%Y A061525 Sequence in context: A142025 A142387 A031433 this_sequence A142496 A142560 A142820
%Y A061525 Adjacent sequences: A061522 A061523 A061524 this_sequence A061526 A061527 A061528
%K A061525 easy,nonn
%O A061525 1,1
%A A061525 Jason Earls (zevi_35711(AT)yahoo.com), May 03 2001
%E A061525 More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
%I A142496
%S A142496 83,389,491,593,797,1103,1307,1409,1511,1613,2531,2633,2837,2939,3041,3347,
%T A142496 3449,4673,4877,5081,5387,5591,5693,5897,6101,6203,6917,7019,7121,7529,
%U A142496 7937,8039,8243,8447,8753,9059,9161,9467,10079,10181,10487,10589,10691
%N A142496 Primes congruent to 32 mod 51.
%Y A142496 Sequence in context: A142387 A031433 A061525 this_sequence A142560 A142820 A142900
%Y A142496 Adjacent sequences: A142493 A142494 A142495 this_sequence A142497 A142498 A142499
%K A142496 nonn
%O A142496 1,1
%A A142496 njas, Jul 11 2008
%I A142560
%S A142560 83,401,613,719,1249,1567,2203,2309,2521,3581,3793,4111,4217,5171,5701,
%T A142560 5807,6337,6761,7079,7927,8563,8669,9199,9623,9941,10259,10789,11213,11743,
%U A142560 12379,12697,13121,13757,14923,15241,15559,15877,16301,16619,16831,16937
%N A142560 Primes congruent to 30 mod 53.
%Y A142560 Sequence in context: A031433 A061525 A142496 this_sequence A142820 A142900 A033248
%Y A142560 Adjacent sequences: A142557 A142558 A142559 this_sequence A142561 A142562 A142563
%K A142560 nonn
%O A142560 1,1
%A A142560 njas, Jul 11 2008
%I A142820
%S A142820 83,449,571,937,1181,1303,1669,1913,2767,3011,3499,4231,4597,5573,5939,
%T A142820 6427,6793,7159,8501,8623,8867,9721,10331,10453,11551,12161,12527,12893,
%U A142820 13259,13381,14479,14723,16187,16553,17041,18749,19237,19603,21067,21433
%N A142820 Primes congruent to 22 mod 61.
%Y A142820 Sequence in context: A061525 A142496 A142560 this_sequence A142900 A033248 A142289
%Y A142820 Adjacent sequences: A142817 A142818 A142819 this_sequence A142821 A142822 A142823
%K A142820 nonn
%O A142820 1,1
%A A142820 njas, Jul 11 2008
%I A142900
%S A142900 83,461,587,839,1091,1217,1721,1847,1973,2099,2351,2477,2729,3359,3863,
%T A142900 3989,4241,4493,4871,5501,5879,6131,6257,6761,7013,7517,7643,8147,8273,
%U A142900 9029,9281,9533,10037,10163,10289,10667,11171,11423,11549,11801,11927,12809
%N A142900 Primes congruent to 20 mod 63.
%Y A142900 Sequence in context: A142496 A142560 A142820 this_sequence A033248 A142289 A142751
%Y A142900 Adjacent sequences: A142897 A142898 A142899 this_sequence A142901 A142902 A142903
%K A142900 nonn
%O A142900 1,1
%A A142900 njas, Jul 11 2008
%I A033248
%S A033248 83,521,691,787,1163,1193,1291,1409,1627,1913,1931,2099,
%T A033248 2273,2347,2579,2689,2713,2833,2897,2939,2953,3067,3209,
%U A033248 3323,3371,3691,3851,3889,4451,4513,4817,5099,5147,5153
%N A033248 Primes of form x^2+74*y^2.
%D A033248 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.
%Y A033248 Sequence in context: A142560 A142820 A142900 this_sequence A142289 A142751 A059236
%Y A033248 Adjacent sequences: A033245 A033246 A033247 this_sequence A033249 A033250 A033251
%K A033248 nonn
%O A033248 1,1
%A A033248 njas
%I A142289
%S A142289 83,599,857,1201,1373,1459,1889,2663,2749,4211,4297,4813,5501,6361,6619,
%T A142289 6791,7307,7393,7823,8081,8167,8597,8941,9199,9371,9629,9887,9973,11177,
%U A142289 11779,12037,12553,12983,13241,13327,13499,13757,15391,15649,15907,17627
%N A142289 Primes congruent to 40 mod 43.
%Y A142289 Sequence in context: A142820 A142900 A033248 this_sequence A142751 A059236 A059935
%Y A142289 Adjacent sequences: A142286 A142287 A142288 this_sequence A142290 A142291 A142292
%K A142289 nonn
%O A142289 1,1
%A A142289 njas, Jul 11 2008
%I A142751
%S A142751 83,673,1381,1499,2089,2207,2797,3623,4567,5039,5393,6101,6337,6691,7517,
%T A142751 7753,8461,8933,10939,11057,11411,12119,12473,13063,13417,14243,14479,14951,
%U A142751 15187,15541,16249,16603,17783,19081,19553,20143,20261,21323,21559,22031
%N A142751 Primes congruent to 24 mod 59.
%Y A142751 Sequence in context: A142900 A033248 A142289 this_sequence A059236 A059935 A069596
%Y A142751 Adjacent sequences: A142748 A142749 A142750 this_sequence A142752 A142753 A142754
%K A142751 nonn
%O A142751 1,1
%A A142751 njas, Jul 11 2008
%I A059236
%S A059236 83,739,821,1231,1559,1723,2297,2543,2707,2789,2953,3527,3691,4019,
%T A059236 5003,5167,5413,5659,5741,5987,6151,6397,6971,7873,8447,8693,9103,9349,
%U A059236 9431,9677,9923,10169,10333,11071,11317,11399,12301,12547,13121,13367
%N A059236 Primes p such that x^41 = 2 has no solution mod p.
%C A059236 Presumably this is also Primes congruent to 1 mod 41. - njas, Jul 11 2008
%C A059236 Complement of A049573 relative to A000040.
%Y A059236 Cf. A000040, A049573.
%Y A059236 Sequence in context: A033248 A142289 A142751 this_sequence A059935 A069596 A112766
%Y A059236 Adjacent sequences: A059233 A059234 A059235 this_sequence A059237 A059238 A059239
%K A059236 easy,nonn
%O A059236 1,1
%A A059236 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 20 2001
%I A059935
%S A059935 1,83,775,46655,46657,93395,140743,279935,279937,280019,280711,326591,
%T A059935 326593,
%U A059935 19916489515870532960258562190639398471599239042185934648024761145811
%N A059935 Fourth step in Goodstein sequences, i.e. g(6) if g(2)=n: write g(5)=A059934(n) in hereditary representation base 5, bump to base 6, then subtract 1 to produce g(6).
%D A059935 Goodstein, R. L. "On the Restricted Ordinal Theorem." J. Symb. Logic 9, 33-41, 1944
%e A059935 a(12) = 280019 since with g(2) = 12 = 2^(2 + 1) + 2^2, we get g(3) = 3^(3 + 1) + 3^3-1 = 107 = 3^(3 + 1) + 2*3^2 + 2*3 + 2, g(4) = 4^(4 + 1) + 2*4^2 + 2*4 + 1 = 1065, g(5) = 5^(5 + 1) + 2*5^2 + 2*5 = 15685, and g(6) = 6^(6 + 1) + 2*6^2 + 6 + 5 = 280019.
%Y A059935 Cf. A056004, A057650, A059933, A059934, A059936.
%Y A059935 Sequence in context: A142289 A142751 A059236 this_sequence A069596 A112766 A128950
%Y A059935 Adjacent sequences: A059932 A059933 A059934 this_sequence A059936 A059937 A059938
%K A059935 nonn
%O A059935 3,2
%A A059935 Henry Bottomley (se16(AT)btinternet.com), Feb 12 2001
%I A069596
%S A069596 83,809,8009,80021,800011,8000009,80000023,800000011,8000000011,
%T A069596 80000000021,800000000047,8000000000009,80000000000027,800000000000017,
%U A069596 8000000000000011,80000000000000011,800000000000000119
%N A069596 Smallest prime in which the n-th significant digit is a 8.
%p A069596 seq(nextprime(8*10^j),j=1..32);
%Y A069596 Cf. A069588, A069589, A069590, A069591, A069592, A069593, A069594, A069595.
%Y A069596 Sequence in context: A142751 A059236 A059935 this_sequence A112766 A128950 A068851
%Y A069596 Adjacent sequences: A069593 A069594 A069595 this_sequence A069597 A069598 A069599
%K A069596 base,nonn
%O A069596 2,1
%A A069596 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 25 2002
%E A069596 More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Mar 28 2002
%I A112766
%S A112766 83,811,8111,839393939,81313131313,8212121212121,83333333,
%T A112766 89191919191919191,8777777777,829292929292929292929,
%U A112766 85151515151515151515151,8313131313131313131313131,811111111111111111111111111,81313131313131313131313131313
%N A112766 Smallest prime of the form 8 followed by n copies of k.
%e A112766 a(3) = 8111, 8 followed by three copies of 1.
%o A112766 (PARI) for(n=1,100,np=1;k=1;while(np,s="8";for(i=1,n,s=concat(s,Str(k)));m=eval(s);if(isprime(m),print1(m",");np=0,k++))) - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
%Y A112766 Sequence in context: A059236 A059935 A069596 this_sequence A128950 A068851 A037069
%Y A112766 Adjacent sequences: A112763 A112764 A112765 this_sequence A112767 A112768 A112769
%K A112766 base,nonn,dumb
%O A112766 1,1
%A A112766 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 02 2006
%E A112766 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
%I A128950
%S A128950 83,829,8286,82858,828571,8285715,82857143,828571429,8285714286,
%T A128950 82857142858,828571428571,8285714285715,82857142857143,828571428571429,
%U A128950 8285714285714286,82857142857142858,828571428571428571
%N A128950 a(n) is equal to the number of positive integers m less than or equal to 10^n such that m is not divisible by the prime 7 and is not divisible by at least one of the primes 2, 3 and 5.
%H A128950 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
%F A128950 f(n)= 10^n-floor(10^n/7)-floor(10^n/30)+floor(10^n/210)
%e A128950 a(6)=828571
%p A128950 f := n->10^n-floor(10^n/7)-floor(10^n/30)+floor(10^n/210);
%Y A128950 Sequence in context: A059935 A069596 A112766 this_sequence A068851 A037069 A103233
%Y A128950 Adjacent sequences: A128947 A128948 A128949 this_sequence A128951 A128952 A128953
%K A128950 nonn
%O A128950 2,1
%A A128950 Milan R. Janjic (agnus(AT)blic.net), Apr 28 2007
%I A068851
%S A068851 83,839,83903,8390303,8390303123
%N A068851 a(1) = 83 ( the smallest prime beginning with 8) and then the smallest prime with leading digits containing a(n-1).
%Y A068851 Cf. A068849, A068850.
%Y A068851 Sequence in context: A069596 A112766 A128950 this_sequence A037069 A103233 A093283
%Y A068851 Adjacent sequences: A068848 A068849 A068850 this_sequence A068852 A068853 A068854
%K A068851 base,nonn
%O A068851 1,1
%A A068851 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 12 2002
%I A037069
%S A037069 83,881,8887,88883,888887,28888883,88888883,888888883,48888888883,
%T A037069 288888888889,888888888887,48888888888883,88888888888889,
%U A037069 888888888888883,18888888888888883,88888888888888889
%N A037069 Smallest prime containing exactly n 8's.
%t A037069 f[n_, b_] := Block[{k = 10^(n + 1), p = Permutations[ Join[ Table[b, {i, 1, n}], {x}]], c = Complement[Table[j, {j, 0, 9}], {b}], q = {}}, Do[q = Append[q, Replace[p, x -> c[[i]], 2]], {i, 1, 9}]; r = Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]; If[r ? Infinity, r, p = Permutations[ Join[ Table[ b, {i, 1, n}], {x, y}]]; q = {}; Do[q = Append[q, Replace[p, {x -> c[[i]], y -> c[[j]]}, 2]], {i, 1, 9}, {j, 1, 9}]; Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]]]; Table[ f[n, 8], {n, 1, 18}]
%Y A037069 Cf. A065591, A037068, A034388, A036507-A036536.
%Y A037069 Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037067, A037071.
%Y A037069 Sequence in context: A112766 A128950 A068851 this_sequence A103233 A093283 A084299
%Y A037069 Adjacent sequences: A037066 A037067 A037068 this_sequence A037070 A037071 A037072
%K A037069 nonn,base,easy
%O A037069 1,1
%A A037069 Patrick De Geest (pdg(AT)worldofnumbers.com), Jan 04 1999.
%E A037069 Corrected by Jud McCranie (j.mccranie(AT)comcast.net), Jan 04 2001. More terms from Erich Friedman (efriedma(AT)stetson.edu), Jun 03 2001.
%I A103233
%S A103233 1,1,83,1779,27691,376772,4767554,57675292,676752609,7767525702,
%T A103233 87675256564,976752565041,10767525649806,117675256497468,
%U A103233 1276752564973977,13767525649738780,147675256497386491
%N A103233 Number of decimal digits in the numerator of BernoulliB[10^n].
%H A103233 Eric Weisstein's World of Mathematics, Bernoulli Number
%e A103233 -1/2, 5/66, -94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330, ...
%Y A103233 Cf. A068399, A114471.
%Y A103233 Sequence in context: A128950 A068851 A037069 this_sequence A093283 A084299 A017799
%Y A103233 Adjacent sequences: A103230 A103231 A103232 this_sequence A103234 A103235 A103236
%K A103233 nonn
%O A103233 0,3
%A A103233 Eric Weisstein (eric(AT)weisstein.com), Jan 26, 2005
%E A103233 More terms from Eric Weisstein (eric(AT)weisstein.com), Feb 20, 2006
%I A093283
%S A093283 0,83,2498,9960,55527888,999984,9999923,4440400499909009,
%T A093283 444040040777070070,9999999941
%N A093283 a(n) is the largest number such that all of a(n)'s length-n substrings are distinct and divisible by 83.
%H A093283 Hans Havermann, Table of A093211-A093299
%e A093283 a(5) is 55527888 because its length-5 substrings (55527, 55278, 52788, 27888) are all distinct and divisible by 83, and there is no larger number with this property
%Y A093283 Cf. A093211, A093212, ..., A093299.
%Y A093283 Sequence in context: A068851 A037069 A103233 this_sequence A084299 A017799 A017746
%Y A093283 Adjacent sequences: A093280 A093281 A093282 this_sequence A093284 A093285 A093286
%K A093283 base,nonn
%O A093283 1,2
%A A093283 Hans Havermann (pxp(AT)rogers.com), Mar 29 2004
%I A084299
%S A084299 83,2903,5897,319499,346943,7974179,15262433,33954251
%N A084299 Smallest primes such that the subsequent terms of consecutive prime differences[A001223] modulo 6 [A054763] displays repeatedly n times a {0,2,4} pattern of remainders of differences.
%e A084299 n=1: a(1)=83 is followed by [6,8,4],
%e A084299 n=2: a(2)=2903 is followed by [6,2,4,18,2,4]
%e A084299 n=3: a(3)=5897 is followed by [6,20,4,12,14,28,6,20,4]
%e A084299 n=4: a(4)=319499 is followed by [12,8,22,6,20,10,12,2,10,6,32,34]
%e A084299 n=5: a(5)=346943 is followed by [18,2,40,....,30,2,10] differences corresponding to n "wavelet" of [0,2,4] remainders modulo 6.
%t A084299 d[x_] := Prime[x+1]-Prime[x] md[x_] := Mod[Prime[x+1]-Prime[x], 6] h={k1=0, k2=2, k3=4}; k=0; Do[If[Equal[md[n], k1]&&Equal[md[n+1], k2]&& Equal[md[n+2], k3]&&Equal[md[n+3], k1]&&Equal[md[n+4], k2]&&Equal[md[n+5], k3] &&Equal[md[n+6], k1]&&Equal[md[n+7], k2]&&Equal[md[n+8], k3] &&Equal[md[n+9], k1]&&Equal[md[n+10], k2]&&Equal[md[n+11], k3]&& Equal[md[n+12], k1]&&Equal[md[n+13], k2]&&Equal[md[n+14], k3], k=k+1; Print[{de, k, n, Prime[n], Table[md[n+j], {j, -1, 15}], Table[d[n+j], {j, -1, 15}]}]], {n, 2, 10000000}]
%Y A084299 Cf. A001223, A054763, A016045.
%Y A084299 Sequence in context: A037069 A103233 A093283 this_sequence A017799 A017746 A097839
%Y A084299 Adjacent sequences: A084296 A084297 A084298 this_sequence A084300 A084301 A084302
%K A084299 more,nonn
%O A084299 1,1
%A A084299 Labos E. (labos(AT)ana.sote.hu), Jun 02 2003
%I A017799
%S A017799 1,83,3403,91881,1837620,29034396,377447148,4151918628,
%T A017799 39443226966,328693558050,2432332329570,16141841823510,
%U A017799 96851050941060,528955739755020,2644778698775100,12165982014365460
%N A017799 Binomial coefficients C(83,n).
%Y A017799 Sequence in context: A103233 A093283 A084299 this_sequence A017746 A097839 A087189
%Y A017799 Adjacent sequences: A017796 A017797 A017798 this_sequence A017800 A017801 A017802
%K A017799 nonn,fini
%O A017799 0,2
%A A017799 njas
%I A017746
%S A017746 1,83,3486,98770,2123555,36949857,541931236,6890268572,
%T A017746 77515521435,783768050065,7210666060598,60962903966874,
%U A017746 477542747740513,3489735464257595,23929614612052080,154744841157936784
%N A017746 Binomial coefficients C(n,82).
%Y A017746 Sequence in context: A093283 A084299 A017799 this_sequence A097839 A087189 A116263
%Y A017746 Adjacent sequences: A017743 A017744 A017745 this_sequence A017747 A017748 A017749
%K A017746 nonn
%O A017746 82,2
%A A017746 njas
%I A097839
%S A097839 1,83,6888,571621,47437655,3936753744,326703123097,27112422463307,
%T A097839 2250004361331384,186723249568041565,15495779709786118511,
%U A097839 1285962992662679794848,106719432611292636853873
%N A097839 Chebyshev polynomials S(n,83).
%C A097839 Used for all positive integer solutions of Pell equation x^2 - 85*y^2 = -4. See A097840 with A097841.
%H A097839 Tanya Khovanova, Recursive Sequences
%H A097839 Index entries for sequences related to Chebyshev polynomials.
%F A097839 a(n)= S(n, 83)=U(n, 83/2)= S(2*n+1, sqrt(85))/sqrt(85) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x).
%F A097839 a(n)=83*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=83; a(-1):=0.
%F A097839 a(n)=(ap^(n+1) - am^(n+1))/(ap-am) with ap := (83+9*sqrt(85))/2 and am := (83-9*sqrt(85))/2 = 1/ap.
%F A097839 G.f.: 1/(1-83*x+x^2).
%Y A097839 Sequence in context: A084299 A017799 A017746 this_sequence A087189 A116263 A087532
%Y A097839 Adjacent sequences: A097836 A097837 A097838 this_sequence A097840 A097841 A097842
%K A097839 nonn,easy
%O A097839 0,2
%A A097839 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 10 2004
%I A087189
%S A087189 1,83,16907,279021,89444018,1695011087,658067703933,5768410509553937,
%T A087189 122108313460051500,1226978854222034501448,593538703869555995238872,
%U A087189 13175226571428140572866093,6594852118968926152838341468
%N A087189 (P(p)-1)/2/p^2 where p runs through the odd primes different from 5 and P(k) is the k-th central pentanomial coefficient (A005191).
%H A087189 M. F. Hasler, Table of n, a(n) for n = 1..100
%o A087189 (PARI) A087189(n) = local(p=prime(n+2-(n==1))); (A005191(p)-1)/2/p^2 \\ - M. F. Hasler
%Y A087189 Sequence in context: A017799 A017746 A097839 this_sequence A116263 A087532 A116307
%Y A087189 Adjacent sequences: A087186 A087187 A087188 this_sequence A087190 A087191 A087192
%K A087189 nonn
%O A087189 1,2
%A A087189 Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 19 2003
%E A087189 Corrected and extended by M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jul 23 2007
%I A116263
%S A116263 83,76980,714687,952311,90438188,96320541,32980078899026,34346653774235,
%T A116263 42816188292270,42881990066485,57118009933509,57183811707724,
%U A116263 65653346225759,67019921100968,81321742742207
%N A116263 n times n+7 gives the concatenation of two numbers m and m-4.
%e A116263 96320541 * 96320548 = 92776472//92776468, where // denotes
%e A116263 concatenation.
%Y A116263 Cf. A116132, A116250, A116262, A116264, A116269.
%Y A116263 Sequence in context: A017746 A097839 A087189 this_sequence A087532 A116307 A093676
%Y A116263 Adjacent sequences: A116260 A116261 A116262 this_sequence A116264 A116265 A116266
%K A116263 nonn,base
%O A116263 1,1
%A A116263 Giovanni Resta (g.resta(AT)iit.cnr.it), Feb 06 2006
%I A087532
%S A087532 83,38383883,88838333,3338388883,3388338883,3388833883,3388838833,
%T A087532 3833388883,3838383883,3838388383,3838838383,3838883383,3883883833,
%U A087532 8333888383,8338383883,8338888333,8383338883,8383388383,8383888333
%N A087532 Primes consisting only of digits 3 and 8 occurring with equal frequency.
%C A087532 There are 18 digit pairs which can produce such primes. (1,0),(7,0),(1,3),(1,4),(1,6),(1,7),(1,9),(2,3),(2,9),(3,4),(3,5),(3,7),(3,8),(4,7),(4,9),(5,9),(6,7),(7,9).
%p A087532 Primes consisting only of digits x and y, occurring with equal frequency. PARI CODE: d1=x; d2=y; k=0; a=vector(100); for(n=1,3000,B=binary(n); L=length(B); s=sum(j=1,length(B),B[j]); if(L%2==0 & s==L/2, C=vector(L,n,(d2-d1)*B[n]+d1); p=subst(Pol(C),x,10); if(isprime(p),if(k<100,k++; a[k]=p)); D=vector(L,n,d2-(d2-d1)*B[n]); q=subst(Pol(D),x,10); if(isprime(q ),if(k<100,k++; a[k]=q))); ); a=vector(k,n,a[n]); vecsort(a)
%Y A087532 Cf. A087510, A087511, A087531.
%Y A087532 Sequence in context: A097839 A087189 A116263 this_sequence A116307 A093676 A130432
%Y A087532 Adjacent sequences: A087529 A087530 A087531 this_sequence A087533 A087534 A087535
%K A087532 base,nonn
%O A087532 0,1
%A A087532 Amarnath Murthy and Paul D. Hanna (pauldhanna(AT)juno.com) (amarnath_murthy(AT)yahoo.com), Sep 12 2003
%I A116307
%S A116307 83,77394229,89158935,36623663376237623663376337,
%T A116307 37633762366336633762366237,62366237633663366237633763,
%U A116307 63376336623762376336623663,86194223018927804587702130
%N A116307 Numbers n such that n times n+1 gives the concatenation of two numbers m and m+3.
%e A116307 89158935 * 89158936 = 79493157//79493160, where // denotes concatenation.
%Y A116307 Cf. A116099, A116301, A116308, A116314.
%Y A116307 Sequence in context: A087189 A116263 A087532 this_sequence A093676 A130432 A066689
%Y A116307 Adjacent sequences: A116304 A116305 A116306 this_sequence A116308 A116309 A116310
%K A116307 nonn,base
%O A116307 1,1
%A A116307 Giovanni Resta (g.resta(AT)iit.cnr.it), Feb 06 2006
%I A093676
%S A093676 83,83333333,833333333333333333333333,833333333333333333333333333333
%N A093676 Primes of the form 8*10^n + 3*R_n, where R_n is the repunit (A002275) of length n.
%Y A093676 Cf. A056723 (corresponding n).
%Y A093676 Sequence in context: A116263 A087532 A116307 this_sequence A130432 A066689 A008898
%Y A093676 Adjacent sequences: A093673 A093674 A093675 this_sequence A093677 A093678 A093679
%K A093676 nonn
%O A093676 1,1
%A A093676 Rick L. Shepherd (rshepherd2(AT)hotmail.com), Apr 08 2004
%I A130432
%S A130432 84,14,36,48,5,72,49,344,9
%N A130432 For digit n from 1 to 9, a(n) = the number of numbers m such that m is equal to the number of n's in the decimal digits of all numbers <= m.
%C A130432 Note: sequences A101639, A101640, and A101641 are defined so that they exclude 0, so they have 13, 35, and 47 elements, respectively. This sequence counts all the zeros, so elements 2,3,4 of this sequence are 14,36,48.
%e A130432 a(3)=36 because there are 36 numbers m such that m is equal to the number of 3's in the decimal digits of all numbers <= m.
%Y A130432 Cf. A014778 for proof these sequences are finite, and also A101639, A101640, A101641, A130427, A130428, A130429, A130430, A130431 for the numbers themselves.
%Y A130432 Sequence in context: A087532 A116307 A093676 this_sequence A066689 A008898 A033404
%Y A130432 Adjacent sequences: A130429 A130430 A130431 this_sequence A130433 A130434 A130435
%K A130432 base,fini,nonn,full
%O A130432 1,1
%A A130432 Graeme McRae (g_m(AT)mcraefamily.com), May 26 2007
%I A066689
%S A066689 84,36,7,7,27,18821,18,9,18,77,9,66,66,9,15488,55,55,62025,9,44,9,1547,
%T A066689 33,11,336,96,11,11,2667,1462,182,11,22,246,22,11,22,143,143,11,11,11,
%U A066689 11,48,117,3762,11,495,117,130,11,104,832,435,11,13,91,91,405,5445
%N A066689 Least number k such that the square root of {k^2 + (Prime[n + k] - Prime[n])^2} is an integer; or 0 if no such number exists.
%C A066689 The square root of {k^2 + (Prime[n + k] - Prime[n])^2} = distance between the points (n,Prime[n]) and (n+k,Prime[n+k]).
%e A066689 k = 84 is the least k such that d[(1,p(1)),(1+k,p(1+k))] = Sqrt[k^2 + (p(1 + k) - p(1))^2] (= 445) is an integer; so a(1) = 84.
%t A066689 a = {}; Do[k = 1; While[ !IntegerQ[ Sqrt[k^2 + (Prime[n + k] - Prime[n])^2]], k++ ]; a = Append[a, k], {n, 1, 60} ]; a
%Y A066689 Sequence in context: A116307 A093676 A130432 this_sequence A008898 A033404 A128873
%Y A066689 Adjacent sequences: A066686 A066687 A066688 this_sequence A066690 A066691 A066692
%K A066689 nonn
%O A066689 1,1
%A A066689 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Jan 11 2002
%E A066689 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 13 2002
%I A008898
%S A008898 84,42,21,62,31,92,46,23,68,34,17,50,25,74,37,110,55,164,
%T A008898 82,41,122,61,182,91,272,136,68,34,17,50,25,74,37,110,55,
%U A008898 164,82,41,122,61,182,91,272,136,68,34,17,50,25,74,37,110
%N A008898 x->x/2 if x even, x->3x-1 if x odd.
%D A008898 R. K. Guy, Unsolved Problems in Number Theory, E16.
%Y A008898 Sequence in context: A093676 A130432 A066689 this_sequence A033404 A128873 A095607
%Y A008898 Adjacent sequences: A008895 A008896 A008897 this_sequence A008899 A008900 A008901
%K A008898 nonn
%O A008898 0,1
%A A008898 njas
%I A033404
%S A033404 84,42,28,21,16,14,12,10,9,8,7,7,6,6,5,5,4,4,4,4,4,3,3,3,3,3,3,3,2,2,2,
%T A033404 2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A033404 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0
%N A033404 [ 84/n ].
%Y A033404 Sequence in context: A130432 A066689 A008898 this_sequence A128873 A095607 A068405
%Y A033404 Adjacent sequences: A033401 A033402 A033403 this_sequence A033405 A033406 A033407
%K A033404 easy,nonn
%O A033404 1,1
%A A033404 Jeff Burch (jmburch(AT)osprey.smcm.edu)
%I A128873
%S A128873 0,1,84,85,1444,1529,85539,87068,172607,259675,691957,1643589,192991870,
%T A128873 194635459,776898247,1748431953,9519058012,11267489965,99658977732,
%U A128873 210585445429,8312491349463,16835568144355,25148059493818
%N A128873 Numerator of the continued fraction convergents of the decimal concatenation of the powers of 2.
%F A128873 The powers of 2 = 1,2,4,8,16,32,64,... are concatenated and then preceded by a decimal point to create the fraction N = .1248163264128... This number is then evaluated with n=0,m=steps to iterate,x = N, a(0)=floor(N) using the loop: do a(n)=floor(x) x=1/(x-a(n)) n=n+1 loop until n=m
%o A128873 (PARI) g(n) = f=".";for(x=0,n,a=concat(f,2^x));f=eval(f) { default(realprecision,1000); cf = vector(1000); cf = contfrac(f); for(m1=0,m-1, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n];); numer=numerator(r); denom=denominator(r); print1(numer","); numer2=numer;denom2=denom; ) }
%Y A128873 Sequence in context: A066689 A008898 A033404 this_sequence A095607 A068405 A045569
%Y A128873 Adjacent sequences: A128870 A128871 A128872 this_sequence A128874 A128875 A128876
%K A128873 frac,nonn
%O A128873 0,3
%A A128873 Cino Hilliard (hillcino368(AT)hotmail.com), Apr 18 2007
%I A095607
%S A095607 84,90,99,108,130,154,213,300,413,1230,11000,1101100,111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
%N A095607 a(n) = 108 written in base 13 - n.
%Y A095607 Sequence in context: A008898 A033404 A128873 this_sequence A068405 A045569 A113931
%Y A095607 Adjacent sequences: A095604 A095605 A095606 this_sequence A095608 A095609 A095610
%K A095607 nonn,fini,full
%O A095607 0,1
%A A095607 njas, Jun 04 2004
%I A068405
%S A068405 84,90,105,110,114,132,140,154,165,182,186,204,220,234,246,252,258,264,
%T A068405 266,273,286,290,294,300,308,318,322,340,345,354,357,364,370,380,385,
%U A068405 402,406,410,414,426,444,450,465,468,470,480,492,504,516,518,525,532
%N A068405 Numbers n such that (n+1) is squarefree and composite, and such that there is one more distinct prime factor in n than in (n+1).
%C A068405 For example 322=2*7*23 has 3 distinct prime factors and 323=17*19 has 2 distinct prime factors, hence 322 is in the sequence.
%F A068405 n such that (1-isprime(n+1))*issquarefree(n+1)*omega(n)=omega(n+1)+1
%Y A068405 Sequence in context: A033404 A128873 A095607 this_sequence A045569 A113931 A111313
%Y A068405 Adjacent sequences: A068402 A068403 A068404 this_sequence A068406 A068407 A068408
%K A068405 easy,nonn
%O A068405 1,1
%A A068405 Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 09 2002
%I A045569
%S A045569 84,91,110,115,181,189,217,224,228,302,308,319,328,353,378,404,407,435,
%T A045569 443,518,527,539,554,589,602,605,630,639,663,668,672,675,676,713,715,
%U A045569 736,757,775,776,792,802,808,819
%N A045569 Numbers n such that final 2 nonzero digits of n! are '92'.
%H A045569 Index entries for sequences related to final digits of numbers
%Y A045569 Sequence in context: A128873 A095607 A068405 this_sequence A113931 A111313 A015708
%Y A045569 Adjacent sequences: A045566 A045567 A045568 this_sequence A045570 A045571 A045572
%K A045569 nonn
%O A045569 1,1
%A A045569 Jeff Burch (gburch(AT)erols.com)
%I A113931
%S A113931 84,99,564
%N A113931 (RSA-1536)-10^n = prime where RSA-1536 is the 463 decimal digit unfactored RSA challenge number.
%C A113931 This sequence shows that the difference between a composite number and a prime rests on the modification of a single decimal digit of the given composite number
%e A113931 (RSA-1536)-10^84 = prime
%t A113931 Position[PrimeQ[Table[ \
%t A113931 184769970321174147430683562020016440301854933866341017147178577491065169671116\
%t A113931 124985933768430543574458561606154457179405222971773252466096064694607124962372\
%t A113931 044202226975675668737842756238950876467844093328515749657884341508847552829818\
%t A113931 672645133986336493190808467199043187438128336350279547028265329780293491615581\
%t A113931 188104984490831954500984839377522725705257859194499387007369575568843693381277\
%t A113931 9613089230392569695253261620823676490316036551371447913932347169566988069 - \
%t A113931 10^n, {n, 1463}]], True]
%Y A113931 Sequence in context: A095607 A068405 A045569 this_sequence A111313 A015708 A114822
%Y A113931 Adjacent sequences: A113928 A113929 A113930 this_sequence A113932 A113933 A113934
%K A113931 nonn,bref
%O A113931 84,1
%A A113931 Joao da Silva (zxawyh66(AT)yahoo.com), Jan 30 2006
%I A111313
%S A111313 84,104,101,32,79,110,45,76,105,110,101,32,69,110,99,121,99,108,111,
%T A111313 112,101,100,105,97,32,111,102,32,73,110,116,101,103,101,114,32,83,
%U A111313 101,113,117,101,110,99,101,115
%N A111313 Character codes of the string "The On-Line Encyclopedia of Integer Sequences".
%t A111313 ToCharacterCode["The On-Line Encyclopedia of Integer Sequences"]
%Y A111313 Sequence in context: A068405 A045569 A113931 this_sequence A015708 A114822 A099637
%Y A111313 Adjacent sequences: A111310 A111311 A111312 this_sequence A111314 A111315 A111316
%K A111313 fini,full,nonn,word,dumb
%O A111313 1,1
%A A111313 Zak Seidov (zakseidov(AT)yahoo.com), Nov 03 2005
%I A015708
%S A015708 1,84,120,162,234,252,270,486,540,588,648,672,819,936,1350,1458,
%T A015708 1536,1550,1638,1764,1782,1800,1872,1920,2268,2325,3000,3042,3240,
%U A015708 3276,3402,3564,3724,3780,4116,4374,4464,4650,4680,4704,5292,5376
%N A015708 Numbers n such that n | (phi(n) * sigma(n)) but (phi(n) + sigma(n))/n does not increase.
%D A015708 R. K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
%Y A015708 Sequence in context: A045569 A113931 A111313 this_sequence A114822 A099637 A101260
%Y A015708 Adjacent sequences: A015705 A015706 A015707 this_sequence A015709 A015710 A015711
%K A015708 nonn
%O A015708 1,2
%A A015708 Robert G. Wilson v (rgwv(AT)rgwv.com)
%I A114822
%S A114822 84,132,156,162,224,234,260,354,364,368,405,434,438,455,472,512,536,584,
%T A114822 585,595,610,615,645,693,741,775,777,822,830,834,867,904,931,957,962
%N A114822 Indices of Fibonacci numbers with 13 prime factors when counted with multiplicity.
%H A114822 Blair Kelly, Fibonacci and Lucas Factorizations.
%e A114822 a(1)=84 because 84th fibonacci number(i.e. 160500643816367088) consists of 13 prime factors (i.e. 2*2*2*2*3*3*13*29*83*211*281*421*1427 )
%o A114822 (PARI) n=1;while(n<360,if(bigomega(fibonacci(n))==13,print1(n,", "));n++)
%Y A114822 Sequence in context: A113931 A111313 A015708 this_sequence A099637 A101260 A039499
%Y A114822 Adjacent sequences: A114819 A114820 A114821 this_sequence A114823 A114824 A114825
%K A114822 hard,more,nonn
%O A114822 1,1
%A A114822 Shyam Sunder Gupta (guptass(AT)rediffmail.com), Feb 19 2006
%E A114822 More terms from Ryan Propper (rpropper(AT)stanford.edu), May 24 2006
%I A099637
%S A099637 84,132,168,228,234,252,260,264,276,308,336,340,372,396,456,468,504,516,
%T A099637 520,528,532,552,558,564,580,588,616,644,672,680,684,702,708,740,744,
%U A099637 756,792,804,820,828,836,852,855,868,884,912,936,948,996,1008,1012,1032
%N A099637 Numbers such that GCD[Sum,n] = A099635 and GCD[Sum,Product] = A099636 are not identical. Sum and Product here are the sum and product of all distinct prime factors of n.
%C A099637 Of the first million integers, 75811 (of which 6300 are odd) belong to this sequence. - Robert G. Wilson v Nov 04 2004. - Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 04 2004
%e A099637 n=84: 84 is here because its factor-list={2,3,7} and sum=2+3+7=12,product=2.3.7=42,GCD[12,84]=12,GCD[12,42]=6. So 6 is not equal 12.
%t A099637 <Cofacient Numbers.
%e A101260 84 is a term of the sequence because 2*2*3*7=84 and 84-42-28-21-14-12-7-6-4-3-2=g[84]=-55
%t A101260 Select[ Range[5500], DivisorSigma[1, # ] == 2# + 56 &] (from Robert G. Wilson v Dec 22 2004)
%Y A101260 Sequence in context: A015708 A114822 A099637 this_sequence A039499 A055712 A066292
%Y A101260 Adjacent sequences: A101257 A101258 A101259 this_sequence A101261 A101262 A101263
%K A101260 nonn,easy
%O A101260 1,1
%A A101260 Vassil K. Tintschev (tinchev(AT)sunhe.jinr.ru), Dec 17 2004
%I A039499
%S A039499 84,151,228,295,372,439,516,583,660,727,804,871,948,1009,1010,1011,
%T A039499 1012,1013,1014,1016,1017,1018,1019,1020,1032,1044,1056,1068,1080,1104,
%U A039499 1116,1128,1140,1159,1236,1303,1380,1447,1524,1591,1668,1747
%N A039499 Numbers n such that representation in base 12 has same nonzero number of 0's and 7's.
%Y A039499 Sequence in context: A114822 A099637 A101260 this_sequence A055712 A066292 A044254
%Y A039499 Adjacent sequences: A039496 A039497 A039498 this_sequence A039500 A039501 A039502
%K A039499 nonn,base,easy
%O A039499 1,1
%A A039499 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A055712
%S A055712 1,84,156,204,364,476,514,1092,1428,2316,2652,2892,6069,6188,6748,
%T A055712 12138,12532,16212,16388,18564,20244,24276,30108,37596,39372,49164,
%U A055712 63291,78897,87724,99202,114716,126582,147679,157794,167331
%N A055712 Numbers n such that n | Sigma_8[n].
%C A055712 sigma_8(n) is the sum of the 8th powers of the divisors of n.
%C A055712 Problem 11090 proves that this sequence is infinite. - T. D. Noe (noe(AT)sspectra.com), Apr 18 2006
%D A055712 Florian Luca, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113 (2006), 372-373.
%t A055712 Do[If[Mod[DivisorSigma[8, n], n]==0, Print[n]], {n, 1, 10000}]
%Y A055712 Sequence in context: A099637 A101260 A039499 this_sequence A066292 A044254 A044635
%Y A055712 Adjacent sequences: A055709 A055710 A055711 this_sequence A055713 A055714 A055715
%K A055712 nonn
%O A055712 1,2
%A A055712 Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 09 2000
%I A066292
%S A066292 1,84,156,364,1092,435708,986076,1118480,1441188,1674036,2446668,
%T A066292 2597868,3108924,3875508,4150692,5537196,6066396,6686316,13729212,
%U A066292 14639436,18735444,23307732,27092052,31806684,58266468,69728724
%N A066292 Numbers n such that n divides sigma_(2^k)(n), the sum of the 2^k powers of the divisors of n, for all k>1.
%C A066292 Let d be the vector of divisors of n. The sequence d^(2^k) mod n has some period p. Thus if n divides sigma_(2^k)(n) for one period, then n divides sigma_(2^k)(n) for all k. For these n, the first period ends for k<158. Hence it is easy to verify divisibility for all k. - T. D. Noe (noe(AT)sspectra.com), Apr 11 2006
%e A066292 n=84 is here because 84 divides each one of sigma_4(n)=53771172, sigma_8(n)=2488859101224132, sigma_16(n)=6144339637187846520573009496452, etc.
%t A066292 t={}; Do[If[Mod[DivisorSigma[4,n],n]==0, AppendTo[t,n]], {n,10^8}]; Do[t=Select[t,Mod[DivisorSigma[2^k,# ],# ]==0&],{k,3,20}]; t - T. D. Noe (noe(AT)sspectra.com), Apr 11 2006
%Y A066292 Cf. A066135, A066284, A066289-A066292.
%Y A066292 Cf. A118076.
%Y A066292 Sequence in context: A101260 A039499 A055712 this_sequence A044254 A044635 A044416
%Y A066292 Adjacent sequences: A066289 A066290 A066291 this_sequence A066293 A066294 A066295
%K A066292 nonn
%O A066292 1,2
%A A066292 Labos E. (labos(AT)ana.sote.hu), Dec 12 2001
%E A066292 Edited by T. D. Noe (noe(AT)sspectra.com), Apr 11 2006
%I A044254
%S A044254 84,165,246,327,408,489,570,651,732,756,813,894,975,1056,1137,
%T A044254 1218,1299,1380,1461,1485,1542,1623,1704,1785,1866,1947,2028,
%U A044254 2109,2190,2214,2271,2352,2433,2514,2595,2676,2757,2838,2919
%N A044254 Numbers n such that string 0,3 occurs in the base 9 representation of n but not of n-1.
%Y A044254 Sequence in context: A039499 A055712 A066292 this_sequence A044635 A044416 A044797
%Y A044254 Adjacent sequences: A044251 A044252 A044253 this_sequence A044255 A044256 A044257
%K A044254 nonn,base
%O A044254 1,1
%A A044254 Clark Kimberling (ck6(AT)evansville.edu)
%I A044635
%S A044635 84,165,246,327,408,489,570,651,732,764,813,894,975,1056,1137,1218,
%T A044635 1299,1380,1461,1493,1542,1623,1704,1785,1866,1947,2028,2109,2190,2222,
%U A044635 2271,2352,2433,2514,2595,2676,2757,2838,2919
%N A044635 Numbers n such that string 0,3 occurs in the base 9 representation of n but not of n+1.
%Y A044635 Sequence in context: A055712 A066292 A044254 this_sequence A044416 A044797 A072589
%Y A044635 Adjacent sequences: A044632 A044633 A044634 this_sequence A044636 A044637 A044638
%K A044635 nonn,base
%O A044635 1,1
%A A044635 Clark Kimberling (ck6(AT)evansville.edu)
%I A044416
%S A044416 84,184,284,384,484,584,684,784,840,884,984,1084,1184,1284,1384,
%T A044416 1484,1584,1684,1784,1840,1884,1984,2084,2184,2284,2384,2484,
%U A044416 2584,2684,2784,2840,2884,2984,3084,3184,3284,3384,3484,3584
%N A044416 Numbers n such that string 8,4 occurs in the base 10 representation of n but not of n-1.
%Y A044416 Sequence in context: A066292 A044254 A044635 this_sequence A044797 A072589 A135804
%Y A044416 Adjacent sequences: A044413 A044414 A044415 this_sequence A044417 A044418 A044419
%K A044416 nonn,base
%O A044416 1,1
%A A044416 Clark Kimberling (ck6(AT)evansville.edu)
%I A044797
%S A044797 84,184,284,384,484,584,684,784,849,884,984,1084,1184,1284,1384,1484,
%T A044797 1584,1684,1784,1849,1884,1984,2084,2184,2284,2384,2484,2584,2684,2784,
%U A044797 2849,2884,2984,3084,3184,3284,3384,3484,3584
%N A044797 Numbers n such that string 8,4 occurs in the base 10 representation of n but not of n+1.
%Y A044797 Sequence in context: A044254 A044635 A044416 this_sequence A072589 A135804 A112066
%Y A044797 Adjacent sequences: A044794 A044795 A044796 this_sequence A044798 A044799 A044800
%K A044797 nonn,base
%O A044797 1,1
%A A044797 Clark Kimberling (ck6(AT)evansville.edu)
%I A072589
%S A072589 1,84,204,294,364,440,444,564,600,644,804,884,950,1000,1134,1164,1204,
%T A072589 1240,1274,1284,1320,1344,1350,1400,1450,1484,1524,1564,1640,1644,1734,
%U A072589 1884,1900,1924,2004,2044,2090,2200,2254,2324,2364,2440,2444,2600,2700
%N A072589 n, Phi(n) and sigma(n) end with the same digit.
%F A072589 It seems that lim n -> infinity a(n)/n exists around 40
%Y A072589 Sequence in context: A044635 A044416 A044797 this_sequence A135804 A112066 A096383
%Y A072589 Adjacent sequences: A072586 A072587 A072588 this_sequence A072590 A072591 A072592
%K A072589 base,nonn
%O A072589 1,2
%A A072589 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2002
%I A135804
%S A135804 1,0,0,84,210,3696,33264,392964,4879875,66106040,963266304,15032793048,
%T A135804 250055167908,4415595820608,82483140014880,1624829831302104,
%U A135804 33659674920420549,731455984834451184,16636624374027720832
%N A135804 Seventh column (k=6) of triangle A134832 (circular succession numbers).
%C A135804 a(n) enumerates circular permutations of {1,2,...,n+6} with exactly six successor pairs (i,i+1). Due to cyclicity also (n+6,1) is a successor pair.
%D A135804 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=6.
%F A135804 a(n)= binomial(n+6,6)*A000757(n), n>=0.
%F A135804 E.g.f.: diff(((x^6)/6!)*(1-ln(1-x))/e^x,x$6).
%e A135804 a(0)=1 because from the 6!/6=120 circular permutations of n=6 elements only one, namely (1,2,3,4,5,6), has six successors.
%Y A135804 Cf. A135803 (column k=5). A135805 (column k=7).
%Y A135804 Sequence in context: A044416 A044797 A072589 this_sequence A112066 A096383 A137210
%Y A135804 Adjacent sequences: A135801 A135802 A135803 this_sequence A135805 A135806 A135807
%K A135804 nonn,easy
%O A135804 0,4
%A A135804 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008
%I A112066
%S A112066 84,239,575,659,839,1175,1320,1344,1764,1835,1955,2099,2160,2375,2459,
%T A112066 2759,2784,2879,3084,3299,3515,3695,3779,3780,3840,3935,4139,4439,4475,
%U A112066 4620,4764,4800,4859,4884,5040,5544,5795,5964,6024,6119,6155
%N A112066 Positive integers i for which A112049(i) == 6.
%Y A112066 Row 6 of A112060.
%Y A112066 Sequence in context: A044797 A072589 A135804 this_sequence A096383 A137210 A131072
%Y A112066 Adjacent sequences: A112063 A112064 A112065 this_sequence A112067 A112068 A112069
%K A112066 nonn
%O A112066 1,1
%A A112066 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A096383
%S A096383 84,240,486,840,1320,1944,2730,3696,4860,6240,7854,9720,11856,14280,
%T A096383 17010,20064,23460,27216,31350,35880,40824,46200,52026,58320,65100,
%U A096383 72384,80190,88536,97440,106920,116994,127680,138996,150960,163590
%N A096383 Area of the Pythagorean triangle a=u^2-v^2 (cf. A096382) when u=3, v=4,4,5...
%C A096383 When u = 1 except for the leading zeros, we get A007531. Since sides a,b of pythagorean triple triangles are of opposite parity, the area will always be an integer.
%F A096383 The area of a Pythagorean triangle of sides a < b < c is A = 1/2*ab. Substituting a=u^2-v^2, b=2uv into A and simplifying, we get A = uv(v^2-u^2).
%F A096383 a(n)=(n-3)*(n*3)*(n+3), n>=3 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 05 2007
%p A096383 seq((n-3)*(n*3)*(n+3), n=3..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 05 2007
%o A096383 (PARI) areagen(n,u) = for(v=u+1,n,print1(u*v*(v^2-u^2)","))
%Y A096383 Cf. A007531.
%Y A096383 Sequence in context: A072589 A135804 A112066 this_sequence A137210 A131072 A083986
%Y A096383 Adjacent sequences: A096380 A096381 A096382 this_sequence A096384 A096385 A096386
%K A096383 nonn
%O A096383 2,1
%A A096383 Cino Hilliard (hillcino368(AT)gmail.com), Aug 05 2004
%I A137210
%S A137210 1,84,252,12717,177744,189264,143747328
%N A137210 Numbers n such that abundance(n) = abundance(sigma(n)).
%e A137210 abundance(84) = sigma(84) - 2(84) = 56 = abundance(224) = abundance(sigma(84)), so 84 is a term in the sequence.
%t A137210 abund[n_] := DivisorSigma[1,n]-2n; l = {}; For[i = 1, i <= 10^6, i++, If[abund[i] == abund[DivisorSigma[1, i]], l = Append[l, i]]]; l
%Y A137210 Sequence in context: A135804 A112066 A096383 this_sequence A131072 A083986 A064198
%Y A137210 Adjacent sequences: A137207 A137208 A137209 this_sequence A137211 A137212 A137213
%K A137210 nonn
%O A137210 1,2
%A A137210 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Mar 05 2008
%E A137210 a(7) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Apr 27 2008
%I A131072
%S A131072 84,336,420,924,1092,1428,1452,1596,1680,1840,1932,2057,2100,2436,2604,
%T A131072 2625,2632,2961,3000,3108,3384,3444,3468,3500,3528,3612,3696,3948,4114,
%U A131072 4368,4452,4500,4620,4956,5124,5250,5376,5460,5520,5544,5628,5712,5808
%N A131072 Numbers m such that A081211(m) <> A081213(m).
%C A131072 Subsequence of A013929;
%C A131072 A081212(a(n)) > 2; A081211(a(n)) <> A081213(a(n));
%C A131072 suggested by Andrew Plewe regarding the equality of initial terms of A081211 and A081213.
%Y A131072 Sequence in context: A112066 A096383 A137210 this_sequence A083986 A064198 A008429
%Y A131072 Adjacent sequences: A131069 A131070 A131071 this_sequence A131073 A131074 A131075
%K A131072 nonn
%O A131072 1,1
%A A131072 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 13 2007
%I A083986
%S A083986 84,403,574,255,976,1207,162,1729,84,736,168,1300,806,216,1148,2668,403,
%T A083986 736,1462,1855,252,2701,3154,1209,574,168,1462,270,1472,3478,336,4606,
%U A083986 255,1300,1855,270,3640,1425,4930,5605,976,806,252,1472,3640,5092,2924
%N A083986 If k is a number with exactly two distinct decimal digits, say a and b, neither of which is 0 (i.e. a member of A101594), define the self-complement of k, SC(k), to be the number obtained by replacing a with b and vice versa. Then a(n) = lcm(A101594(n), SC(A101594(n))).
%e A083986 a(7) = lcm(18, 81) = 162.
%Y A083986 Cf. A083983, A083984, A083985.
%Y A083986 Cf. A101594.
%Y A083986 Sequence in context: A096383 A137210 A131072 this_sequence A064198 A008429 A069080
%Y A083986 Adjacent sequences: A083983 A083984 A083985 this_sequence A083987 A083988 A083989
%K A083986 base,nonn
%O A083986 0,1
%A A083986 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 22 2003
%E A083986 Corrected and extended by David Wasserman (wasserma(AT)spawar.navy.mil), Dec 07 2004
%I A064198
%S A064198 0,0,84,468,1476,3540,7200,13104,22008,34776,52380,75900,106524,
%T A064198 145548,194376,254520,327600,415344,519588,642276,785460,951300,
%U A064198 1142064,1360128,1607976,1888200,2203500,2556684,2950668,3388476
%N A064198 3*(n-2)*(n-3)*(3*n^2-3*n-8)/2.
%D A064198 L. Berzolari, Allgemeine Theorie der Ho"heren Ebenen Algebraischen Kurven, Encyclopa"die der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B.G. Teubner, 1906. p. 341.
%Y A064198 Sequence in context: A137210 A131072 A083986 this_sequence A008429 A069080 A093284
%Y A064198 Adjacent sequences: A064195 A064196 A064197 this_sequence A064199 A064200 A064201
%K A064198 nonn
%O A064198 2,3
%A A064198 Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Sep 22 2001
%I A008429
%S A008429 1,84,574,1288,3444,4424,9240,11088,18494,19740,34440,31304,52808,52248,
%T A008429 74048,71120,110964,94864,145222,132888,181384,163856,249480,201040,
%U A008429 295960,264684,346696,314272,454608,352520,518336,452256,591934,517216
%N A008429 Theta series of D_7 lattice.
%D A008429 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
%F A008429 G.f.: (theta_3(q^(1/2))^7+theta_4(q^(1/2))^7)/2
%e A008429 1 + 84*q^2 + 574*q^4 + 1288*q^6 + 3444*q^8 + ...
%o A008429 (PARI) {a(n)=if(n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^7, n))} /* Michael Somos Nov 03 2006 */
%Y A008429 A008451(2n)=a(n).
%Y A008429 Sequence in context: A131072 A083986 A064198 this_sequence A069080 A093284 A027794
%Y A008429 Adjacent sequences: A008426 A008427 A008428 this_sequence A008430 A008431 A008432
%K A008429 nonn,easy
%O A008429 0,2
%A A008429 njas
%I A069080
%S A069080 84,864,3300,8736,18900,35904,62244,100800,154836,228000,324324,448224,
%T A069080 604500,798336,1035300,1321344,1662804,2066400,2539236,3088800,3722964,
%U A069080 4449984,5278500,6217536,7276500,8465184,9793764,11272800,12913236
%N A069080 (2n+1)(2n+2)(2n+6)(2n+7).
%D A069080 Konrad Knopp, Theory and application of infinite series, Dover, p. 268
%H A069080 Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
%F A069080 sum(n=1, inf, (-1)^n/a(n))=(Pi-149/60)/60
%Y A069080 Sequence in context: A083986 A064198 A008429 this_sequence A093284 A027794 A027821
%Y A069080 Adjacent sequences: A069077 A069078 A069079 this_sequence A069081 A069082 A069083
%K A069080 easy,nonn
%O A069080 0,1
%A A069080 Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002
%I A093284
%S A093284 0,84,924,21008400,99960,88880488880,50505008080800,99999984,999999924,
%T A093284 22222221008888888400
%N A093284 a(n) is the largest number such that all of a(n)'s length-n substrings are distinct and divisible by 84.
%H A093284 Hans Havermann, Table of A093211-A093299
%e A093284 a(4) is 21008400 because its length-4 substrings (2100, 1008, 0084, 0840, 8400) are all distinct and divisible by 84, and there is no larger number with this property
%Y A093284 Cf. A093211, A093212, ..., A093299.
%Y A093284 Sequence in context: A064198 A008429 A069080 this_sequence A027794 A027821 A092719
%Y A093284 Adjacent sequences: A093281 A093282 A093283 this_sequence A093285 A093286 A093287
%K A093284 base,nonn
%O A093284 1,2
%A A093284 Hans Havermann (pxp(AT)rogers.com), Mar 29 2004
%I A027794
%S A027794 84,960,5940,26400,94380,288288,780780,1921920,4375800,9335040,
%T A027794 18845112,36279360,67016040,119380800,205931880,345181056,563861100,
%U A027794 899870400,1406047500,2154952800,3244861620,4807202400
%N A027794 12*(n+1)*C(n+3,9).
%F A027794 Number of 13-subsequences of [ 1, n ] with just 3 contiguous pairs; g.f. 12*(7+3x)/(1-x)^11
%F A027794 C(n+7, 7)*C(n+9, 3) - Zerinvary Lajos (zlaja(AT)freemail.hu), May 10 2005
%Y A027794 Sequence in context: A008429 A069080 A093284 this_sequence A027821 A092719 A105942
%Y A027794 Adjacent sequences: A027791 A027792 A027793 this_sequence A027795 A027796 A027797
%K A027794 nonn
%O A027794 0,1
%A A027794 thi ngoc dinh (via rkg(AT)cpsc.ucalgary.ca)
%I A027821
%S A027821 84,1050,6930,32340,120120,378378,1051050,2642640,6126120,13273260,
%T A027821 27159132,52907400,98760480,177578940,308897820,521694096,858049500,
%U A027821 1377926550,2165313150,3336032700,5047562520
%N A027821 21*(n+1)*C(n+6,9).
%F A027821 Number of 16-subsequences of [ 1, n ] with just 6 contiguous pairs; g.f. 42*(2+3x)/(1-x)^11
%F A027821 C(n+4, 4)*C(n+9, 6) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 08 2005
%Y A027821 Sequence in context: A069080 A093284 A027794 this_sequence A092719 A105942 A008359
%Y A027821 Adjacent sequences: A027818 A027819 A027820 this_sequence A027822 A027823 A027824
%K A027821 nonn
%O A027821 0,1
%A A027821 thi ngoc dinh (via rkg(AT)cpsc.ucalgary.ca)
%I A092719
%S A092719 0,0,0,0,0,0,84,1176,8736
%N A092719 Disk degeneracies for brane III in the O(K)->P^1 x P^1 geometry.
%H A092719 M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web
%Y A092719 Sequence in context: A093284 A027794 A027821 this_sequence A105942 A008359 A098935
%Y A092719 Adjacent sequences: A092716 A092717 A092718 this_sequence A092720 A092721 A092722
%K A092719 nonn
%O A092719 0,7
%A A092719 Sam Alexander (amnalexander(AT)yahoo.com), Mar 05 2004
%I A105942
%S A105942 84,1470,12936,76616,360360,1387386,4624620,13741728,37165128,929128820
%N A105942 C(n+6,n)*C(n+9,6)
%e A105942 If n=0 then C(0+6,0)*C(0+9,6)= C(6,0)*C(9,6)=1*84=84
%e A105942 If n=6 then C(6+6,6)*C(6+9,6)= C(12,6)*C(15,6)=924*5005=4624620
%Y A105942 Cf. A062145.
%Y A105942 Sequence in context: A027794 A027821 A092719 this_sequence A008359 A098935 A104674
%Y A105942 Adjacent sequences: A105939 A105940 A105941 this_sequence A105943 A105944 A105945
%K A105942 easy,nonn
%O A105942 0,1
%A A105942 Zerinvary Lajos (zlaja(AT)freemail.hu), Apr 27 2005
%I A008359
%S A008359 1,84,1498,11620,55650,195972,559258,1371316,2999682,
%T A008359 6003956,11193882,19695172,33023074,53163684,82663002,
%U A008359 124723732,183309826,263258772,370401626,511690788,695335522
%N A008359 Coordination sequence for D_7 lattice.
%D A008359 R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H A008359 T. D. Noe, Table of n, a(n) for n=0..1000
%H A008359 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps).
%p A008359 484/45*n^6+392/9*n^4+1246/45*n^2+2;
%Y A008359 A row of array A103903.
%Y A008359 Sequence in context: A027821 A092719 A105942 this_sequence A098935 A104674 A140903
%Y A008359 Adjacent sequences: A008356 A008357 A008358 this_sequence A008360 A008361 A008362
%K A008359 nonn
%O A008359 0,2
%A A008359 njas and J. H. Conway (conway(AT)math.princeton.edu)
%I A098935
%S A098935 84,1908,74850,806344,19324668,62374858,408913614,891388420,3398376408,
%T A098935 17229339930,27483954208,94862481154,194638347078,271671521308,
%U A098935 506393669424,1174292792658,2487936351660,3141898009018,6059361174964
%N A098935 p^7 - p^5 - p^3 - p^2 where p is prime.
%H A098935 Dario Alejandro Alpern, Factorization using Elliptic Curve Method.
%e A098935 If p=2, 2^7 - 2^5 - 2^3 - 2^2 = 84
%t A098935 Table[p = Prime[n]; p^7 - p^5 - p^3 - p^2, {n, 20}]
%Y A098935 Sequence in context: A092719 A105942 A008359 this_sequence A104674 A140903 A026809
%Y A098935 Adjacent sequences: A098932 A098933 A098934 this_sequence A098936 A098937 A098938
%K A098935 nonn
%O A098935 1,1
%A A098935 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Oct 20 2004
%E A098935 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 14 2004
%I A104674
%S A104674 1,84,2184,30576,286650,2018016,11435424,54609984,259397424
%N A104674 C(n+6,6)*C(n+11,n+0)
%e A104674 If n=0 then C(0+6,6)*C(0+11,0+0)= C(6,6)*C(11,0)=1*1=1
%e A104674 If n=8 then C(8+6,6)*C(8+11,8+0)= C(14,6)*C(19,8)=3432*75582=259397424
%Y A104674 Cf. A062190.
%Y A104674 Sequence in context: A105942 A008359 A098935 this_sequence A140903 A026809 A114253
%Y A104674 Adjacent sequences: A104671 A104672 A104673 this_sequence A104675 A104676 A104677
%K A104674 easy,nonn
%O A104674 0,2
%A A104674 Zerinvary Lajos (zlaja(AT)freemail.hu), Apr 22 2005
%I A140903
%S A140903 1,84,2520,41580,457380,3737448,24293412,131589315,614083470,2530768240,
%T A140903 9386849472,31803696288,99604982880,291153026880,800670823920,2085276513474,
%U A140903 5172303508911,12276881393700,27999904933000,61578738292500,130994770549500,270273795363000
%N A140903 Number of 3 X 6 matrices with elements in 0..n with each row and each column in nondecreasing order. 3,6,n can be permuted, see formula.
%F A140903 (empirical) set p,q,r to n,6,3 (in any order) in s=p+q+r-1; a(n) = product {i in 0..r-1} (binomial(s,p+i)*i!/(s-i)^(r-i-1))
%Y A140903 Sequence in context: A008359 A098935 A104674 this_sequence A026809 A114253 A017800
%Y A140903 Adjacent sequences: A140900 A140901 A140902 this_sequence A140904 A140905 A140906
%K A140903 nonn
%O A140903 0,2
%A A140903 Ron Hardin rhh(at)cadence.com, Jul 05 2008
%I A026809
%S A026809 0,1,84,2925,85320,2362041,64304604,1741001445,47050068240,1270739210481,
%T A026809 34313445309924,926494403955165,25015631334908760,675424587904113321,
%U A026809 18236486750190760044,492385348146244352085,13294406252968671566880
%N A026809 3^n*(3^n-1)*(3^n-2)/6.
%F A026809 a(n)=C(3^n,3),n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008
%p A026809 seq(binomial(3^n,3),n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008
%Y A026809 Sequence in context: A098935 A104674 A140903 this_sequence A114253 A017800 A035737
%Y A026809 Adjacent sequences: A026806 A026807 A026808 this_sequence A026810 A026811 A026812
%K A026809 nonn
%O A026809 0,3
%A A026809 njas
%I A114253
%S A114253 1,84,3276,92400,2187900,46558512,923410488,17439488352,317907339750,
%T A114253 5644249611000,98209943231400,1682207622669600,28457345616827400
%N A114253 C(5+2*n,5+n)*C(10+2*n,0+n)
%e A114253 If n=1 then C(5+2*1,5+1)*C(10+2*1,0+1)=C7,6)*C(12,1)=7*12=84
%e A114253 If n=11 then C(5+2*n,5+n)*C(10+2*n,0+n)=C(27,16)*C(32,11)=13037895*129024480=1682207622669600
%Y A114253 Cf. A062190.
%Y A114253 Sequence in context: A104674 A140903 A026809 this_sequence A017800 A035737 A035806
%Y A114253 Adjacent sequences: A114250 A114251 A114252 this_sequence A114254 A114255 A114256
%K A114253 easy,nonn
%O A114253 0,2
%A A114253 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 04 2006
%I A017800
%S A017800 1,84,3486,95284,1929501,30872016,406481544,4529365776,
%T A017800 43595145594,368136785016,2761025887620,18574174153080,
%U A017800 112992892764570,625806790696080,3173734438530120,14810760713140560
%N A017800 Binomial coefficients C(84,n).
%Y A017800 Sequence in context: A140903 A026809 A114253 this_sequence A035737 A035806 A017747
%Y A017800 Adjacent sequences: A017797 A017798 A017799 this_sequence A017801 A017802 A017803
%K A017800 nonn,fini
%O A017800 0,2
%A A017800 njas
%I A035737
%S A035737 1,84,3528,98812,2076816,34949796,490681688,5913144396,
%T A035737 62456027424,587522034932,4985149915368,38549117382300,
%U A035737 273998113272240,1803067831236420,11053262513080440,63460928860322028
%N A035737 Coordination sequence for 42-dimensional cubic lattice.
%C A035737 Is this the same as A035806? - Andrew Plewe, May 02 2007
%D A035737 J. Serra-Sagrista, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44.
%H A035737 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps).
%F A035737 ((1+x)/(1-x))^42.
%Y A035737 Sequence in context: A026809 A114253 A017800 this_sequence A035806 A017747 A004379
%Y A035737 Adjacent sequences: A035734 A035735 A035736 this_sequence A035738 A035739 A035740
%K A035737 nonn,easy
%O A035737 0,2
%A A035737 Serra-Sagrista, Joan; jserra(AT)ccd.uab.es
%E A035737 Recomputed Nov 25 1998 by njas.
%I A035806
%S A035806 1,84,3528,98812,2076816,34949796,490681688,5913144396,
%T A035806 62456027424,587522034932,4985149915368,38549117382300,
%U A035806 273998113272240,1803067831236420,11053262513080440,63460928860322028
%N A035806 Coordination sequence for lattice D*_42 (with edges defined by l_1 norm = 1).
%C A035806 Is this the same as A035737? - Andrew Plewe, May 02 2007
%D A035806 J. Serra-Sagrista, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44.
%F A035806 a(m)=add(2^k*binomial(n, k)*binomial(m-1, k-1), k=0..n)+2^n*binomial((n+2*m)/2-1, n-1); with n=42.
%Y A035806 Sequence in context: A114253 A017800 A035737 this_sequence A017747 A004379 A075906
%Y A035806 Adjacent sequences: A035803 A035804 A035805 this_sequence A035807 A035808 A035809
%K A035806 nonn
%O A035806 0,2
%A A035806 njas, J. Serra-Sagrista (jserra(AT)ccd.uab.es)
%I A017747
%S A017747 1,84,3570,102340,2225895,39175752,581106988,7471375560,
%T A017747 84986896995,868754947060,8079421007658,69042324974532,
%U A017747 546585072715045,4036320536972640,27965935149024720,182710776306961504
%N A017747 Binomial coefficients C(n,83).
%Y A017747 Sequence in context: A017800 A035737 A035806 this_sequence A004379 A075906 A075909
%Y A017747 Adjacent sequences: A017744 A017745 A017746 this_sequence A017748 A017749 A017750
%K A017747 nonn
%O A017747 83,2
%A A017747 njas
%I A004379
%S A004379 1,84,4095,152096,4780230,134153712,3470108187,84431259000,
%T A004379 1959267085776,43790762164380,949517708685546,20082459351180240,
%U A004379 416047580521136200,8470484585302467168,169921950043809807675
%N A004379 Binomial coefficient C(7n,n-11).
%D A004379 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%H A004379 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%Y A004379 Sequence in context: A035737 A035806 A017747 this_sequence A075906 A075909 A132052
%Y A004379 Adjacent sequences: A004376 A004377 A004378 this_sequence A004380 A004381 A004382
%K A004379 nonn,easy
%O A004379 11,2
%A A004379 njas
%I A075906
%S A075906 1,84,4158,158760,5182947,152457228,4166544096,107883135360,
%T A075906 2681751885813,64597295294532,1518037879508514,34979886546859800,
%U A075906 793401360863472999,17766424516726033596,393690756719422620612
%N A075906 Seventh column of triangle A075498.
%C A075906 The e.g.f. given below is sum(A075513(7,m)*exp(3*(m+1)*x),m=0..6)/6!.
%F A075906 a(n)=A075498(n+7, 7)=(3^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
%F A075906 a(n)= sum(A075513(7, m)*((m+1)*3)^n, m=0..6)/6!.
%F A075906 G.f.: 1/product(1-3*k*x, k=1..7).
%F A075906 E.g.f.: diff((((exp(3*x)-1)/3)^7)/7!, x$7) = (exp(3*x)-384*exp(6*x)+10935*exp(9*x)-81920*exp(12*x)+234375*exp(15*x)-279936*exp(18*x)+117649*exp(21*x))/6!.
%Y A075906 Cf. A075516.
%Y A075906 Sequence in context: A035806 A017747 A004379 this_sequence A075909 A132052 A097840
%Y A075906 Adjacent sequences: A075903 A075904 A075905 this_sequence A075907 A075908 A075909
%K A075906 nonn,easy
%O A075906 0,2
%A A075906 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002
%I A075909
%S A075909 1,84,4256,169344,5843712,183794688,5421678592,152720375808,
%T A075909 4157366140928,110282217357312,2867778350481408,73424436820180992,
%U A075909 1857023919127527424,46511918954689069056,1155904251854380335104
%N A075909 Sixth column of triangle A075499.
%C A075909 The e.g.f. given below is sum(A075513(6,m)*exp(4*(m+1)*x),m=0..5)/5!.
%F A075909 a(n)=A075499(n+6, 6)=(4^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
%F A075909 a(n)= sum(A075513(6, m)*((m+1)*4)^n, m=0..5)/5!.
%F A075909 G.f.: 1/product(1-4*k*x, k=1..6).
%F A075909 E.g.f.: diff((((exp(4*x)-1)/4)^6)/6!, x$6) = (-exp(4*x)+160*exp(8*x)-2430*exp(12*x)+10240*exp(16*x)-15625*exp(20*x)+7776*exp(24*x))/5!.
%Y A075909 Cf. A075908, A075910.
%Y A075909 Sequence in context: A017747 A004379 A075906 this_sequence A132052 A097840 A076230
%Y A075909 Adjacent sequences: A075906 A075907 A075908 this_sequence A075910 A075911 A075912
%K A075909 nonn,easy
%O A075909 0,2
%A A075909 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002
%I A132052
%S A132052 1,84,4662,220500,9740115,419625360,18048090060,785470565880,
%T A132052 34872721208325,1587323312675100,74301594199682850,3583275362669702700,
%U A132052 178220792065162821975,9146316814629741747000,484394828691800237211000
%N A132052 Seventh column of triangle A035342.
%C A132052 a(n), n>=7, enumerates unordered forests composed of seven plane increasing ternary trees with n vertices. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.
%F A132052 E.g.f. ((x*c(x/2)*(1-2*x)^(-1/2))^7)/7!, where c(x) = g.f. for Catalan numbers A000108, a(0) := 0.
%F A132052 E.g.f. (-1+(1-2*x)^(-1/2))^7/7!.
%e A132052 a(8)=84=3*binomial(8,2) increasing ternary 7-forest with n=8 vertices: there are three 7-forests (six one vertex trees together with any of the three different 2-vertex trees) each with binomial(8,2)= 28 increasing labelings.
%Y A132052 Cf. A132051 (sixth column).
%Y A132052 Sequence in context: A004379 A075906 A075909 this_sequence A097840 A076230 A118076
%Y A132052 Adjacent sequences: A132049 A132050 A132051 this_sequence A132053 A132054 A132055
%K A132052 nonn,easy
%O A132052 7,2
%A A132052 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 14 2007
%I A097840
%S A097840 1,84,6971,578509,48009276,3984191399,330639876841,27439125586404,
%T A097840 2277116783794691,188973253929372949,15682502959354160076,
%U A097840 1301458772372465913359,108005395603955316648721
%N A097840 Chebyshev polynomials S(n,83) + S(n-1,83) with diophantine property.
%C A097840 (9*a(n))^2 - 85*b(n)^2 = -4 with b(n)=A097841(n) give all positive solutions of this Pell equation.
%H A097840 Tanya Khovanova, Recursive Sequences
%H A097840 Index entries for sequences relate d to Chebyshev polynomials.
%F A097840 a(n)= S(n, 83) + S(n-1, 83) = S(2*n, sqrt(85)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 83)=A097839(n).
%F A097840 a(n)= (-2/9)*I*((-1)^n)*T(2*n+1, 9*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
%F A097840 G.f.: (1+x)/(1-83*x+x^2).
%e A097840 All positive solutions of Pell equation x^2 - 85*y^2 = -4 are
%e A097840 (9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...
%Y A097840 Sequence in context: A075906 A075909 A132052 this_sequence A076230 A118076 A056746
%Y A097840 Adjacent sequences: A097837 A097838 A097839 this_sequence A097841 A097842 A097843
%K A097840 nonn,easy
%O A097840 0,2
%A A097840 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 10 2004
%I A076230
%S A076230 1,84,435708,986076,1441188,6066396,7407036,16763292,18735444,78863148
%N A076230 Numbers n such that Sigma[2, n]/n and Sigma[4, n]/n are integers.
%e A076230 n=6066396, sigma[2, n]/n=9156979, sigma[4, n]/n=241153415598179286943
%Y A076230 Intersection of A046762(n) and A046764(n).
%Y A076230 Sequence in context: A075909 A132052 A097840 this_sequence A118076 A056746 A111194
%Y A076230 Adjacent sequences: A076227 A076228 A076229 this_sequence A076231 A076232 A076233
%K A076230 nonn
%O A076230 1,2
%A A076230 Labos E. (labos(AT)ana.sote.hu), Oct 03 2002
%E A076230 More terms from T. D. Noe (noe(AT)sspectra.com), Apr 11 2006
%I A118076
%S A118076 1,84,435708,986076,1441188,6066396,18735444,78863148
%N A118076 Numbers n such that n divides sigma_(2^k)(n), the sum of the 2^k powers of the divisors of n, for all k>0.
%C A118076 Although these numbers have been tested up to k=20, it is conjectured that n divides sigma_(2^k)(n) for all k>0. Intersection of A046762 and A066292.
%C A118076 Let d be the vector of divisors of n. The sequence d^(2^k) mod n has some period p. Thus if n divides sigma_(2^k)(n) for one period, then n divides sigma_(2^k)(n) for all k. For these n, the first period ends for k<14. Hence it is easy to verify divisibility for all k. Intersection of A046762 and A066292. - T. D. Noe (noe(AT)sspectra.com), Apr 12 2006
%e A118076 n=84 is here because 84 divides each one of sigma_4(n)=53771172, sigma_8(n)=2488859101224132, sigma_16(n)=6144339637187846520573009496452, etc.
%t A118076 t={}; Do[If[Mod[DivisorSigma[2,n],n]==0, AppendTo[t,n]], {n,10^8}]; Do[t=Select[t,Mod[DivisorSigma[2^k,# ],# ]==0&],{k,2,20}]; t
%Y A118076 Cf. A076230 (n divides sigma_2(n) and sigma_4(n)).
%Y A118076 Sequence in context: A132052 A097840 A076230 this_sequence A056746 A111194 A096132
%Y A118076 Adjacent sequences: A118073 A118074 A118075 this_sequence A118077 A118078 A118079
%K A118076 nonn
%O A118076 1,2
%A A118076 T. D. Noe (noe(AT)sspectra.com), Apr 11 2006
%I A056746
%S A056746 0,0,0,84,438984,2194477729800
%N A056746 Raw solutions to Hi-Q puzzle with n holes on a side, any initial peg removed.
%H A056746 Eric Weisstein's World of Mathematics, Hi-Q
%Y A056746 Sequence in context: A097840 A076230 A118076 this_sequence A111194 A096132 A078083
%Y A056746 Adjacent sequences: A056743 A056744 A056745 this_sequence A056747 A056748 A056749
%K A056746 hard,nonn
%O A056746 1,4
%A A056746 David W. Wilson (davidwwilson(AT)comcast.net), Aug 15 2000
%I A111194
%S A111194 1,1,84,1397520,5314794912000,4855173934730716800000,
%T A111194 1090093558153665322315192780800000,
%U A111194 60907190511553979457004412118419080463155200000
%N A111194 Permanent of the inverse Hilbert matrix.
%t A111194 Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m . v), Times @@ v]]; f[n_] := Block[{i = Inverse[Table[1/(i + j - 1), {i, n}, {j, n}]]}, Permanent[i]]; Table[ f[n], {n, 7}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 24 2005)
%o A111194 J (www.jsoftware.com)
%o A111194 H =: % @: >: @: (+/~) @: i. @ x:
%o A111194 perm=: +/ .*
%o A111194 perm@%.@H n
%Y A111194 Cf. A005249 = determinant of inverse Hilbert matrix; and A092326 = (permanent/determinant) of inverse Hilbert matrix.
%Y A111194 Sequence in context: A076230 A118076 A056746 this_sequence A096132 A078083 A105328
%Y A111194 Adjacent sequences: A111191 A111192 A111193 this_sequence A111195 A111196 A111197
%K A111194 nonn
%O A111194 0,3
%A A111194 Roger Hui (RHui000(AT)Shaw.CA), Oct 22 2005
%I A096132
%S A096132 1,1,1,1,84,4686825,1,12870,3284214703056,10078751602022313874633200,1,
%T A096132 3268760,9064807833193439800,25006639164538285144538957539300707000,
%U A096132 137658555538877668586244095134027016988748997970545868021484500,1
%N A096132 Triangle read by rows in which the r-th term of the n-th row is C(n^r,r*n), where r = 1 to n.
%e A096132 1
%e A096132 1 1
%e A096132 1 84 4686825
%e A096132 1 12870 3284214703056 =C(256,16) 10078751602022313874633200
%e A096132 1 3268760 9064807833193439800 25006639164538285144538957539300707000 ...
%e A096132 ...
%t A096132 Flatten[ Table[ Binomial[n^r, r*n], {n, 6}, {r, n}]] (from Robert G. Wilson v Jul 08 2004)
%Y A096132 Cf. A096130, A096131.
%Y A096132 Sequence in context: A118076 A056746 A111194 this_sequence A078083 A105328 A033405
%Y A096132 Adjacent sequences: A096129 A096130 A096131 this_sequence A096133 A096134 A096135
%K A096132 nonn,tabl
%O A096132 1,5
%A A096132 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 04 2004
%E A096132 Edited, corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 08 2004
%I A078083
%S A078083 0,1,85,1,96,1,11,1,2,6,3,1,1,3,1,1,3,2,1,18,1,2,1,7,1,2,1,1,2,2,1,
%T A078083 228,1,11,1,2,2,1,8,1,82,1,1,1,4,11,1,1,1,1,3,1,3,1,3,4,1,1,4,5,8,2,
%U A078083 1,1,6,2,1,1,2,1,2,1,3,15,2,4,1,43,3,3,4,1,4,72,54,1,1,1,70,1,80,1
%N A078083 Continued fraction for constant defined in A065468.
%Y A078083 Sequence in context: A056746 A111194 A096132 this_sequence A105328 A033405 A003906
%Y A078083 Adjacent sequences: A078080 A078081 A078082 this_sequence A078084 A078085 A078086
%K A078083 nonn,cofr
%O A078083 1,3
%A A078083 Benoit Cloitre, Dec 02 2002
%I A105328
%S A105328 85,39,73,42,22,67,35,67,6,54,63,55,8,69,54,65,74,49,50,34,88,85,35,76,
%T A105328 51,14,96,18,79,60,11,30,17,92,28,61,11,57,33,8,7,57,25,63,86,97,10,47,
%U A105328 39,43,91,37,74,94,25,11,67,74,67,64,63,21,18,75,90,69,60,23,99,6,18,36
%N A105328 Digital expansion of e*pi: numbers from each pair of successive digits.
%t A105328 Table[FromDigits[Partition[RealDigits[N[e*Pi, 200]][[1]], 2][[i]]], {i, 100}]
%Y A105328 Sequence in context: A111194 A096132 A078083 this_sequence A033405 A003906 A020312
%Y A105328 Adjacent sequences: A105325 A105326 A105327 this_sequence A105329 A105330 A105331
%K A105328 nonn,base,dumb
%O A105328 1,1
%A A105328 Zak Seidov (zakseidov(AT)yahoo.com), Apr 30 2005
%I A033405
%S A033405 85,42,28,21,17,14,12,10,9,8,7,7,6,6,5,5,5,4,4,4,4,3,3,3,3,3,3,3,2,2,2,
%T A033405 2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A033405 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0
%N A033405 [ 85/n ].
%Y A033405 Sequence in context: A096132 A078083 A105328 this_sequence A003906 A020312 A044978
%Y A033405 Adjacent sequences: A033402 A033403 A033404 this_sequence A033406 A033407 A033408
%K A033405 easy,nonn
%O A033405 1,1
%A A033405 Jeff Burch (jmburch(AT)osprey.smcm.edu)
%I A003906
%S A003906 1,85,85,323,323,324,646,646,816,1140,1215,1215,1615,1920,
%T A003906 1920,1920,1938,1938,2432,2754,3078
%N A003906 Degrees of irreducible representations of Janko group J3.
%D A003906 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%o A003906 (GAP) Display(CharacterTable("J3"));
%Y A003906 Sequence in context: A078083 A105328 A033405 this_sequence A020312 A044978 A095593
%Y A003906 Adjacent sequences: A003903 A003904 A003905 this_sequence A003907 A003908 A003909
%K A003906 nonn,fini,full
%O A003906 1,2
%A A003906 njas
%I A020312
%S A020312 85,87,595,3091,7397,7483,12403,12673,19951,22015,22591,32215,33419,
%T A020312 39391,39865,45141,56641,60451,67861,69751,75055,88831,90865,98413,99269,
%U A020312 102173,112529,113401,118405,134863,163189,188057,188551,220951,229633
%N A020312 Strong pseudoprimes to base 86.
%H A020312 Index entries for sequences related to pseudoprimes
%Y A020312 Sequence in context: A105328 A033405 A003906 this_sequence A044978 A095593 A039487
%Y A020312 Adjacent sequences: A020309 A020310 A020311 this_sequence A020313 A020314 A020315
%K A020312 nonn
%O A020312 1,1
%A A020312 David W. Wilson (davidwwilson(AT)comcast.net)
%I A044978
%S A044978 85,91,93,109,111,117,257,259,265,275,277,281,285,289,291,301,307,309,
%T A044978 329,331,335,339,343,345,353,357,369,379,381,387,409,415,417,433,435,
%U A044978 441,499,517,523,525,571,577,579,595,597,603
%N A044978 Numbers n with property that in base 3 representation the numbers of 0's and 1's are 2 and 3, respectively.
%Y A044978 Sequence in context: A033405 A003906 A020312 this_sequence A095593 A039487 A066474
%Y A044978 Adjacent sequences: A044975 A044976 A044977 this_sequence A044979 A044980 A044981
%K A044978 nonn,base
%O A044978 1,1
%A A044978 Clark Kimberling (ck6(AT)evansville.edu)
%I A095593
%S A095593 85,92,101,122,145,203,245,401,1211,10202,1100101,11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
%N A095593 a(n) = 101 written in base 12 - n.
%Y A095593 Sequence in context: A003906 A020312 A044978 this_sequence A039487 A066474 A027453
%Y A095593 Adjacent sequences: A095590 A095591 A095592 this_sequence A095594 A095595 A095596
%K A095593 nonn,fini,full
%O A095593 0,1
%A A095593 njas, Jun 04 2004
%I A039487
%S A039487 85,95,206,216,327,337,448,458,569,579,690,700,811,821,855,866,877,888,
%T A039487 899,910,921,935,936,937,938,939,940,941,944,945,954,965,975,986,997,
%U A039487 1008,1019,1030,1041,1045,1046,1047,1048,1049,1050,1051,1054,1055
%N A039487 Numbers n such that representation in base 11 has same nonzero number of 7's and 8's.
%Y A039487 Sequence in context: A020312 A044978 A095593 this_sequence A066474 A027453 A029471
%Y A039487 Adjacent sequences: A039484 A039485 A039486 this_sequence A039488 A039489 A039490
%K A039487 nonn,base,easy
%O A039487 1,1
%A A039487 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A066474
%S A066474 85,98,112,113,200,256,265,312,364,400,420,441,481,484,544,625,729,761,
%T A066474 800,924,925,1152,1200,1444,1681,1764,1849,1860,1861,1936,2116,2209,
%U A066474 2245,2664,3364,3481,3721,3844,4704,5101,5304,5476,5724,6400,6889,7321
%N A066474 Numbers having just eight anti-divisors.
%C A066474 See A066272 for definition of anti-divisor.
%H A066474 Jon Perry, The Anti-Divisor
%H A066474 Jon Perry, The Anti-divisor [Cached copy]
%H A066474 Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]
%t A066474 antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2*n], OddQ[ # ] && # != 1 &]]] }, # < n & ]]; Select[ Range[10^4], Length[ antid[ # ]] == 8 & ]
%Y A066474 Cf. A066272.
%Y A066474 Sequence in context: A044978 A095593 A039487 this_sequence A027453 A029471 A083750
%Y A066474 Adjacent sequences: A066471 A066472 A066473 this_sequence A066475 A066476 A066477
%K A066474 nonn
%O A066474 1,1
%A A066474 Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 02 2002
%I A027453
%S A027453 85,115,1489,919,30676,89125,609625,479269,46805584,38580444,
%T A027453 5467738500,4651663500,4005934416,13944602529,3539108614041,
%U A027453 3132881376625,1008229438216000,904461877983664
%N A027453 Third diagonal of A027447.
%F A027453 Numerators of sequence a[ n, n-2 ] in (a[ i, j ])^3 where a[ i, j ] = 1/i if j<=i, 0 if j>i
%Y A027453 Sequence in context: A095593 A039487 A066474 this_sequence A029471 A083750 A043684
%Y A027453 Adjacent sequences: A027450 A027451 A027452 this_sequence A027454 A027455 A027456
%K A027453 nonn
%O A027453 3,1
%A A027453 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A029471
%S A029471 1,85,145,245,1189,356717
%N A029471 Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 2 (most significant digit on left).
%C A029471 The next term is > 410000. - Larry Reeves, Jan 16, 2002
%H A029471 Index entries for related sequences
%Y A029471 Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978
%Y A029471 Sequence in context: A039487 A066474 A027453 this_sequence A083750 A043684 A043574
%Y A029471 Adjacent sequences: A029468 A029469 A029470 this_sequence A029472 A029473 A029474
%K A029471 nonn,base
%O A029471 1,2
%A A029471 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%E A029471 Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12, 2002
%E A029471 One more term from Larry Reeves (larryr(AT)acm.org), Dec 03 2001
%I A083750
%S A083750 85,145,304,915
%N A083750 Numbers n such that fp(n,1111) is prime (cf. A083677).
%C A083750 a(3)=304 because fp(304,1111)=211113111151111...199911112003 is a prime related to prime year 2003; this prime number has 2231 digits. fp(915,1111)=211113111151111...712911117151 is a prime with 7119 digits (prime(915)=7151).
%H A083750 Farideh Firoozbakht, On Solution of puzzle 208 .
%e A083750 a(1)=85 because fp(85,1111)= 211113111151111...4331111439 is prime and fp(k,1111) is composite for k< 85 (prime(85)=439).
%Y A083750 Cf. A083677, A082549.
%Y A083750 Sequence in context: A066474 A027453 A029471 this_sequence A043684 A043574 A043739
%Y A083750 Adjacent sequences: A083747 A083748 A083749 this_sequence A083751 A083752 A083753
%K A083750 nonn,uned
%O A083750 1,1
%A A083750 Farideh Firoozbakht (f.firoozbakht(AT)sci.ui.ac.ir), Jun 17 2003
%I A043684
%S A043684 85,149,165,169,170,171,173,181,213,277,293,297,298,299,301,309,
%T A043684 325,329,330,331,333,337,338,339,340,342,343,345,346,347,349,
%U A043684 357,361,362,363,365,373,405,421,425,426,427,429,437,469,533
%N A043684 (1/2)(n-th number whose base 2 representation has exactly 8 runs).
%Y A043684 Sequence in context: A027453 A029471 A083750 this_sequence A043574 A043739 A043753
%Y A043684 Adjacent sequences: A043681 A043682 A043683 this_sequence A043685 A043686 A043687
%K A043684 nonn,base
%O A043684 1,1
%A A043684 Clark Kimberling (ck6(AT)evansville.edu)
%I A043574
%S A043574 85,149,165,169,171,173,181,213,277,293,297,299,301,309,325,329,
%T A043574 331,333,337,339,343,345,347,349,357,361,363,365,373,405,421,
%U A043574 425,427,429,437,469,533,549,553,555,557,565,581,585,587,589
%N A043574 Numbers n such that base 2 representation has exactly 7 runs.
%Y A043574 Sequence in context: A029471 A083750 A043684 this_sequence A043739 A043753 A043761
%Y A043574 Adjacent sequences: A043571 A043572 A043573 this_sequence A043575 A043576 A043577
%K A043574 nonn,base
%O A043574 1,1
%A A043574 Clark Kimberling (ck6(AT)evansville.edu)
%I A043739
%S A043739 85,149,165,169,171,173,181,213,277,293,297,299,301,309,325,329,
%T A043739 331,333,337,339,343,345,347,349,357,361,363,365,373,405,421,
%U A043739 425,427,429,437,469,533,549,553,555,557,565,581,585,587,589
%N A043739 Number of runs in the base 2 representation of n is congruent to 0 mod 7.
%Y A043739 Sequence in context: A083750 A043684 A043574 this_sequence A043753 A043761 A043770
%Y A043739 Adjacent sequences: A043736 A043737 A043738 this_sequence A043740 A043741 A043742
%K A043739 nonn,base
%O A043739 1,1
%A A043739 Clark Kimberling (ck6(AT)evansville.edu)
%I A043753
%S A043753 85,149,165,169,171,173,181,213,277,293,297,299,301,309,325,329,
%T A043753 331,333,337,339,343,345,347,349,357,361,363,365,373,405,421,
%U A043753 425,427,429,437,469,533,549,553,555,557,565,581,585,587,589
%N A043753 Number of runs in the base 2 representation of n is congruent to 7 mod 8.
%Y A043753 Sequence in context: A043684 A043574 A043739 this_sequence A043761 A043770 A051992
%Y A043753 Adjacent sequences: A043750 A043751 A043752 this_sequence A043754 A043755 A043756
%K A043753 nonn,base
%O A043753 1,1
%A A043753 Clark Kimberling (ck6(AT)evansville.edu)
%I A043761
%S A043761 85,149,165,169,171,173,181,213,277,293,297,299,301,309,325,329,
%T A043761 331,333,337,339,343,345,347,349,357,361,363,365,373,405,421,
%U A043761 425,427,429,437,469,533,549,553,555,557,565,581,585,587,589
%N A043761 Number of runs in the base 2 representation of n is congruent to 7 mod 9.
%Y A043761 Sequence in context: A043574 A043739 A043753 this_sequence A043770 A051992 A044255
%Y A043761 Adjacent sequences: A043758 A043759 A043760 this_sequence A043762 A043763 A043764
%K A043761 nonn,base
%O A043761 1,1
%A A043761 Clark Kimberling (ck6(AT)evansville.edu)
%I A043770
%S A043770 85,149,165,169,171,173,181,213,277,293,297,299,301,309,325,329,
%T A043770 331,333,337,339,343,345,347,349,357,361,363,365,373,405,421,
%U A043770 425,427,429,437,469,533,549,553,555,557,565,581,585,587,589
%N A043770 Number of runs in the base 2 representation of n is congruent to 7 mod 10.
%Y A043770 Sequence in context: A043739 A043753 A043761 this_sequence A051992 A044255 A044636
%Y A043770 Adjacent sequences: A043767 A043768 A043769 this_sequence A043771 A043772 A043773
%K A043770 nonn,base
%O A043770 1,1
%A A043770 Clark Kimberling (ck6(AT)evansville.edu)
%I A051992
%S A051992 85,165,205,221,285,357,365,429,533,629,645,741,957,965,1085,1205,
%T A051992 1245,1365,1469,1517,1533,1685,1853,1965,2013,2037,2045,2085,2373,
%U A051992 2397,2405,2613,2805,2813,3005,3045,3237,3485,3885,3965,4245,4277
%N A051992 Discriminants of real quadratic fields of ERD-type with class groups of exponent 2 and discriminants of the form D = r^2*k^2+4k, k odd.
%C A051992 Excludes discriminants appearing in A051990.
%D A051992 R. A. Mollin, Quadratics, CRC Press, 1996, Appendix A, Table A4.
%Y A051992 Cf. A051990-A051998.
%Y A051992 Sequence in context: A043753 A043761 A043770 this_sequence A044255 A044636 A089060
%Y A051992 Adjacent sequences: A051989 A051990 A051991 this_sequence A051993 A051994 A051995
%K A051992 nonn,fini
%O A051992 0,1
%A A051992 njas, Jan 04 2000
%I A044255
%S A044255 85,166,247,328,409,490,571,652,733,765,814,895,976,1057,1138,
%T A044255 1219,1300,1381,1462,1494,1543,1624,1705,1786,1867,1948,2029,
%U A044255 2110,2191,2223,2272,2353,2434,2515,2596,2677,2758,2839,2920
%N A044255 Numbers n such that string 0,4 occurs in the base 9 representation of n but not of n-1.
%Y A044255 Sequence in context: A043761 A043770 A051992 this_sequence A044636 A089060 A037979
%Y A044255 Adjacent sequences: A044252 A044253 A044254 this_sequence A044256 A044257 A044258
%K A044255 nonn,base
%O A044255 1,1
%A A044255 Clark Kimberling (ck6(AT)evansville.edu)
%I A044636
%S A044636 85,166,247,328,409,490,571,652,733,773,814,895,976,1057,1138,1219,
%T A044636 1300,1381,1462,1502,1543,1624,1705,1786,1867,1948,2029,2110,2191,2231,
%U A044636 2272,2353,2434,2515,2596,2677,2758,2839,2920
%N A044636 Numbers n such that string 0,4 occurs in the base 9 representation of n but not of n+1.
%Y A044636 Sequence in context: A043770 A051992 A044255 this_sequence A089060 A037979 A044417
%Y A044636 Adjacent sequences: A044633 A044634 A044635 this_sequence A044637 A044638 A044639
%K A044636 nonn,base
%O A044636 1,1
%A A044636 Clark Kimberling (ck6(AT)evansville.edu)
%I A089060
%S A089060 85,170,184,217,255,340,368,425,434,510,552,564,651,680,736,765,781,820,
%T A089060 850,868,920,935,1020,1085,1104,1105,1128,1261,1264,1275,1302,1360,1445,
%U A089060 1472,1530,1562,1615,1640,1656,1692,1700,1736,1840,1870,1953,1955,2024
%N A089060 x = non-multiple of 7 such that xy/(x+y) is an integer and the hypotenuse of the right triangle with legs x and y is an integer.
%e A089060 x=85,y=204, 85^2+204^2 = 221^2
%o A089060 (PARI) xydivxpynodiv7(n) = { for(x=1,n, for(y=x,n, h=x*y/(x+y); if(h==floor(h), z = sqrt(x^2+y^2); if(z==floor(z) && floor(z)%7, print1(x",") ) ) ) ) }
%Y A089060 Sequence in context: A051992 A044255 A044636 this_sequence A037979 A044417 A044798
%Y A089060 Adjacent sequences: A089057 A089058 A089059 this_sequence A089061 A089062 A089063
%K A089060 nonn
%O A089060 1,1
%A A089060 Cino Hilliard (hillcino368(AT)gmail.com), Dec 02 2003
%I A037979
%S A037979 85,170,255,256,340,342,343,426,511,512,597,680,681,683,767,768,
%T A037979 853,938,1020,1021,1022,1025,1026,1027,1109,1194,1279,1280,1360,
%U A037979 1361,1362,1363,1368,1369,1370,1371,1372,1373,1374,1375,1450
%N A037979 n-th number whose maximal base 4 run length is 4.
%Y A037979 Sequence in context: A044255 A044636 A089060 this_sequence A044417 A044798 A072289
%Y A037979 Adjacent sequences: A037976 A037977 A037978 this_sequence A037980 A037981 A037982
%K A037979 nonn,base
%O A037979 1,1
%A A037979 Clark Kimberling (ck6(AT)evansville.edu)
%I A044417
%S A044417 85,185,285,385,485,585,685,785,850,885,985,1085,1185,1285,1385,
%T A044417 1485,1585,1685,1785,1850,1885,1985,2085,2185,2285,2385,2485,
%U A044417 2585,2685,2785,2850,2885,2985,3085,3185,3285,3385,3485,3585
%N A044417 Numbers n such that string 8,5 occurs in the base 10 representation of n but not of n-1.
%Y A044417 Sequence in context: A044636 A089060 A037979 this_sequence A044798 A072289 A027524
%Y A044417 Adjacent sequences: A044414 A044415 A044416 this_sequence A044418 A044419 A044420
%K A044417 nonn,base
%O A044417 1,1
%A A044417 Clark Kimberling (ck6(AT)evansville.edu)
%I A044798
%S A044798 85,185,285,385,485,585,685,785,859,885,985,1085,1185,1285,1385,1485,
%T A044798 1585,1685,1785,1859,1885,1985,2085,2185,2285,2385,2485,2585,2685,2785,
%U A044798 2859,2885,2985,3085,3185,3285,3385,3485,3585
%N A044798 Numbers n such that string 8,5 occurs in the base 10 representation of n but not of n+1.
%Y A044798 Sequence in context: A089060 A037979 A044417 this_sequence A072289 A027524 A043340
%Y A044798 Adjacent sequences: A044795 A044796 A044797 this_sequence A044799 A044800 A044801
%K A044798 nonn,base
%O A044798 1,1
%A A044798 Clark Kimberling (ck6(AT)evansville.edu)
%I A072289
%S A072289 1,85,230,1054,205,5405,6510,18615,27335,45034,44556,22660,152889,89531,
%T A072289 181220,53430,221595,304265,246380,720291,360910,595884,811954,1444915,
%U A072289 1362295,40630,2504645,1304445,3311396,2385474,3647810,2420665,1641809
%N A072289 One eighty-fourth the area of primitive Pythagorean triangles with (increasing) square hypotenuses (precisely those of A008846).
%C A072289 For Pythagorean triples (x, y, z) satisfying x^2 + y^2 = z^2, we have 3 and 4 dividing either of x or y, and 7 dividing x, y or (x^2 - y^2), so that 3*4*7 always divide x*y*(x^2 - y^2); if (x, y) be themselves the generators of another Pythagorean triple, (x^2 - y^2, 2*x*y, x^2 + y^2=z^2), the corresponding primitive Pythagorean triangle has area x*y*(x^2 - y^2) and is hence divisible by 84.
%Y A072289 Cf. A020882.
%Y A072289 Sequence in context: A037979 A044417 A044798 this_sequence A027524 A043340 A045129
%Y A072289 Adjacent sequences: A072286 A072287 A072288 this_sequence A072290 A072291 A072292
%K A072289 nonn
%O A072289 1,2
%A A072289 Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 11 2002
%E A072289 Corrected and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 28 2003
%I A027524
%S A027524 85,253,10882,346596,512612,226576,118286,57899,498278,261725,
%T A027524 747515925,66272157632,153348270754
%N A027524 Third diagonal of A027517.
%F A027524 Numerators of sequence a[ n, n-2 ] in (a[ i, j ])^3 where a[ i, j ] = T(i, j)/Partitions(i) if j<=i, 0 if j>i
%Y A027524 Sequence in context: A044417 A044798 A072289 this_sequence A043340 A045129 A045105
%Y A027524 Adjacent sequences: A027521 A027522 A027523 this_sequence A027525 A027526 A027527
%K A027524 nonn
%O A027524 3,1
%A A027524 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A043340
%S A043340 85,277,325,337,340,342,343,345,349,357,373,405,469,597,853,1045,
%T A043340 1093,1105,1108,1110,1111,1113,1117,1125,1141,1173,1237,1285,
%U A043340 1297,1300,1302,1303,1305,1309,1317,1333,1345,1348,1350,1351
%N A043340 Numbers n such that number of 1's in base 4 is 4.
%Y A043340 Sequence in context: A044798 A072289 A027524 this_sequence A045129 A045105 A038472
%Y A043340 Adjacent sequences: A043337 A043338 A043339 this_sequence A043341 A043342 A043343
%K A043340 nonn,base
%O A043340 1,1
%A A043340 Clark Kimberling (ck6(AT)evansville.edu)
%I A045129
%S A045129 85,277,325,337,340,342,345,357,405,597,1045,1093,1105,1108,1110,1113,
%T A045129 1125,1173,1285,1297,1300,1302,1305,1317,1345,1348,1350,1353,1360,1362,
%U A045129 1368,1370,1377,1380,1382,1385,1413,1425,1428
%N A045129 Numbers n with property that in base 4 representation the numbers of 1's and 3's are 4 and 0, respectively.
%Y A045129 Sequence in context: A072289 A027524 A043340 this_sequence A045105 A038472 A020200
%Y A045129 Adjacent sequences: A045126 A045127 A045128 this_sequence A045130 A045131 A045132
%K A045129 nonn,base
%O A045129 1,1
%A A045129 Clark Kimberling (ck6(AT)evansville.edu)
%I A045105
%S A045105 85,277,325,337,340,343,349,373,469,853,1045,1093,1105,1108,1111,1117,
%T A045105 1141,1237,1285,1297,1300,1303,1309,1333,1345,1348,1351,1357,1360,1363,
%U A045105 1372,1375,1393,1396,1399,1405,1477,1489,1492
%N A045105 Numbers n with property that in base 4 representation the numbers of 1's and 2's are 4 and 0, respectively.
%Y A045105 Sequence in context: A027524 A043340 A045129 this_sequence A038472 A020200 A020298
%Y A045105 Adjacent sequences: A045102 A045103 A045104 this_sequence A045106 A045107 A045108
%K A045105 nonn,base
%O A045105 1,1
%A A045105 Clark Kimberling (ck6(AT)evansville.edu)
%I A038472
%S A038472 85,277,325,337,340,1045,1093,1105,1108,1285,1297,1300,1345,1348,1360,
%T A038472 4117,4165,4177,4180,4357,4369,4372,4417,4420,4432,5125,5137,5140,5185,
%U A038472 5188,5200,5377,5380,5392,5440,16405,16453,16465,16468,16645
%N A038472 Sums of 4 distinct powers of 4.
%Y A038472 Base 4 interpretation of A038446.
%Y A038472 Sequence in context: A043340 A045129 A045105 this_sequence A020200 A020298 A068559
%Y A038472 Adjacent sequences: A038469 A038470 A038471 this_sequence A038473 A038474 A038475
%K A038472 nonn,easy
%O A038472 0,1
%A A038472 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A020200
%S A020200 85,305,365,451,511,781,793,949,1037,1105,1241,1387,1541,1729,2465,2485,
%T A020200 2501,2701,2821,2911,4381,4411,4453,5183,5185,5257,6205,6601,6697,8449,
%U A020200 8911,9061,10585,11305,13213,13981,14111,15841,16441,17803,18721,19345
%N A020200 Pseudoprimes to base 72.
%H A020200 Index entries for sequences related to pseudoprimes
%Y A020200 Sequence in context: A045129 A045105 A038472 this_sequence A020298 A068559 A045017
%Y A020200 Adjacent sequences: A020197 A020198 A020199 this_sequence A020201 A020202 A020203
%K A020200 nonn
%O A020200 1,1
%A A020200 David W. Wilson (davidwwilson(AT)comcast.net)
%I A020298
%S A020298 85,305,451,793,1037,1105,1387,2465,4381,5185,5257,6697,14111,19669,
%T A020298 20557,22177,23281,28471,28645,29341,30073,45449,46313,49141,55969,60551,
%U A020298 61249,64345,67405,68251,70801,79729,85285,90751,91001,96049,97147
%N A020298 Strong pseudoprimes to base 72.
%H A020298 Index entries for sequences related to pseudoprimes
%Y A020298 Sequence in context: A045105 A038472 A020200 this_sequence A068559 A045017 A020212
%Y A020298 Adjacent sequences: A020295 A020296 A020297 this_sequence A020299 A020300 A020301
%K A020298 nonn
%O A020298 1,1
%A A020298 David W. Wilson (davidwwilson(AT)comcast.net)
%I A068559
%S A068559 1,85,333,436,1542,1875,2907,3285,3488,3796,5196,10280,17532,17776,
%T A068559 20080,21250,28305,30368,30555,32708,34748,35308,36860,37060,41544,
%U A068559 41568,43068,44004,45162,48468,51930,81324,98304,98688,104856,131070
%N A068559 Numbers n such that phi(n)=tau(n)^3.
%Y A068559 Sequence in context: A038472 A020200 A020298 this_sequence A045017 A020212 A069308
%Y A068559 Adjacent sequences: A068556 A068557 A068558 this_sequence A068560 A068561 A068562
%K A068559 easy,nonn
%O A068559 1,2
%A A068559 Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 25 2002
%I A045017
%S A045017 85,342,343,345,349,357,373,405,469,597,853,1370,1371,1374,1375,1382,
%T A045017 1383,1385,1389,1398,1399,1401,1405,1430,1431,1433,1437,1445,1461,1494,
%U A045017 1495,1497,1501,1509,1525,1622,1623,1625,1629
%N A045017 Numbers n with property that in base 4 representation the numbers of 0's and 1's are 0 and 4, respectively.
%Y A045017 Sequence in context: A020200 A020298 A068559 this_sequence A020212 A069308 A020239
%Y A045017 Adjacent sequences: A045014 A045015 A045016 this_sequence A045018 A045019 A045020
%K A045017 nonn,base
%O A045017 1,1
%A A045017 Clark Kimberling (ck6(AT)evansville.edu)
%I A020212
%S A020212 85,415,481,703,1105,1111,1411,1615,2465,2501,2509,2981,3145,3655,3667,
%T A020212 4141,5713,6161,6533,6973,7055,7141,7201,7885,8401,8695,9061,10585,11441,
%U A020212 13019,13579,13981,14023,14383,14491,15181,15251,15355,15521,16405,16745
%N A020212 Pseudoprimes to base 84.
%H A020212 Index entries for sequences related to pseudoprimes
%Y A020212 Sequence in context: A020298 A068559 A045017 this_sequence A069308 A020239 A008360
%Y A020212 Adjacent sequences: A020209 A020210 A020211 this_sequence A020213 A020214 A020215
%K A020212 nonn
%O A020212 1,1
%A A020212 David W. Wilson (davidwwilson(AT)comcast.net)
%I A069308
%S A069308 85,926,10269,114344,1275129,14226710,158755329,1771648672,19771329973,
%T A069308 220646655318,2462407613981,27480393415984,306680449108553,
%U A069308 3422545943800702,38195526542871473,426261120804626008
%N A069308 Number of 4 X n binary arrays with a path of adjacent 1's from upper left corner to anywhere in right hand column.
%Y A069308 Cf. 2 X n A069306, 3 X n A069307, 5 X n A069309, 6 X n A069310, 7 X n A069311, 8 X n A069312, 9 X n A069313, 10 X n A069314, 11 X n A069315, 12 X n A069316, 13 X n A069317, 14 X n A069318, 15 X n A069319, 16 X n A069320, by columns A069294-A069305.
%Y A069308 Sequence in context: A068559 A045017 A020212 this_sequence A020239 A008360 A020310
%Y A069308 Adjacent sequences: A069305 A069306 A069307 this_sequence A069309 A069310 A069311
%K A069308 nonn
%O A069308 2,1
%A A069308 Ron Hardin (rhh(AT)cadence.com), Mar 14 2002
%I A020239
%S A020239 85,1099,5149,7107,8911,9637,13019,14491,17803,19757,20881,22177,23521,
%T A020239 26521,35371,44173,45629,54097,56033,57205,75241,83333,85285,86347,
%U A020239 102719,110309,153401,184339,191959,222529,242845,253021,253927,269861
%N A020239 Strong pseudoprimes to base 13.
%H A020239 Index entries for sequences related to pseudoprimes
%Y A020239 Sequence in context: A045017 A020212 A069308 this_sequence A008360 A020310 A076463
%Y A020239 Adjacent sequences: A020236 A020237 A020238 this_sequence A020240 A020241 A020242
%K A020239 nonn
%O A020239 1,1
%A A020239 David W. Wilson (davidwwilson(AT)comcast.net)
%I A008360
%S A008360 1,85,1583,13203,68853,264825,824083,2195399,5195081,
%T A008360 11199037,22392919,42088091,75111165,128274849,210937851,
%U A008360 335661583,518971409,782230181,1152631807,1664322595
%N A008360 Crystal ball sequence for D_7 lattice.
%H A008360 Index entries for crystal ball sequences
%H A008360 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps).
%p A008360 1/315 * (2*n+1)*(242*n^6+726*n^5+1856*n^4+2502*n^3+2207*n^2+1077*n+315);
%Y A008360 Sequence in context: A020212 A069308 A020239 this_sequence A020310 A076463 A017801
%Y A008360 Adjacent sequences: A008357 A008358 A008359 this_sequence A008361 A008362 A008363
%K A008360 nonn
%O A008360 0,2
%A A008360 njas and J. H. Conway (conway(AT)math.princeton.edu)
%I A020310
%S A020310 85,1615,2501,2981,5713,6973,7141,8401,13019,14023,14383,15251,26569,
%T A020310 28645,31195,36311,38323,48133,56033,68251,69751,76049,79381,82513,88357,
%U A020310 88831,91741,97567,97621,102331,104653,105001,129397,130561,136729
%N A020310 Strong pseudoprimes to base 84.
%H A020310 Index entries for sequences related to pseudoprimes
%Y A020310 Sequence in context: A069308 A020239 A008360 this_sequence A076463 A017801 A017748
%Y A020310 Adjacent sequences: A020307 A020308 A020309 this_sequence A020311 A020312 A020313
%K A020310 nonn
%O A020310 1,1
%A A020310 David W. Wilson (davidwwilson(AT)comcast.net)
%I A076463
%S A076463 85,2803,24026,115270,397655,1107505,2653588,5685996,11176665,20511535,
%T A076463 35594350,58962098,93912091,144640685,216393640,315628120,450186333,
%U A076463 629480811,864691330,1168973470,1557678815,2048586793,2662148156
%N A076463 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly four ways.
%D A076463 Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.
%F A076463 1/6*n*(n+1)*(97*n^4+146*n^3+30*n^2-19*n+1). G.f.: x*(181*x^4+2976*x^3+6190*x^2+2208*x+85)/((1+x)*(1-x)^7).
%p A076463 seq(1/6*n*(n+1)*(97*n^4+146*n^3+30*n^2-19*n+1),n=1..30);
%Y A076463 Cf. A076389, A076460-A076465.
%Y A076463 Sequence in context: A020239 A008360 A020310 this_sequence A017801 A017748 A093285
%Y A076463 Adjacent sequences: A076460 A076461 A076462 this_sequence A076464 A076465 A076466
%K A076463 easy,nonn
%O A076463 1,1
%A A076463 Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 13 2002
%I A017801
%S A017801 1,85,3570,98770,2024785,32801517,437353560,4935847320,
%T A017801 48124511370,411731930610,3129162672636,21335200040700,
%U A017801 131567066917650,738799683460650,3799541229226200,17984495151670680
%N A017801 Binomial coefficients C(85,n).
%Y A017801 Sequence in context: A008360 A020310 A076463 this_sequence A017748 A093285 A011813
%Y A017801 Adjacent sequences: A017798 A017799 A017800 this_sequence A017802 A017803 A017804
%K A017801 nonn,fini
%O A017801 0,2
%A A017801 njas
%I A017748
%S A017748 1,85,3655,105995,2331890,41507642,622614630,8093990190,
%T A017748 93080887185,961835834245,9041256841903,78083581816435,
%U A017748 624668654531480,4660989191504120,32626924340528840,215337700647490344
%N A017748 Binomial coefficients C(n,84).
%Y A017748 Sequence in context: A020310 A076463 A017801 this_sequence A093285 A011813 A006106
%Y A017748 Adjacent sequences: A017745 A017746 A017747 this_sequence A017749 A017750 A017751
%K A017748 nonn
%O A017748 84,2
%A A017748 njas
%I A093285
%S A093285 0,85,4255,9945,99960,999940,11101005550500,99999950,888800800555500500,
%T A093285 9999999930
%N A093285 a(n) is the largest number such that all of a(n)'s length-n substrings are distinct and divisible by 85.
%H A093285 Hans Havermann, Table of A093211-A093299
%e A093285 a(7) is 11101005550500 because its length-7 substrings (1110100, 1101005, 1010055, 0100555, 1005550, 0055505, 0555050, 5550500) are all distinct and divisible by 85, and there is no larger number with this property
%Y A093285 Cf. A093211, A093212, ..., A093299.
%Y A093285 Sequence in context: A076463 A017801 A017748 this_sequence A011813 A006106 A015338
%Y A093285 Adjacent sequences: A093282 A093283 A093284 this_sequence A093286 A093287 A093288
%K A093285 base,nonn
%O A093285 1,2
%A A093285 Hans Havermann (pxp(AT)rogers.com), Mar 29 2004
%I A011813
%S A011813 0,0,0,0,0,1,85,4699,219836,9595906,409867615,17589965248,
%T A011813 770635739545,34813976611381,1632431803786770,79803825438459069,
%U A011813 4080007508711841376,218627996809373217066,12298853131687073056568
%N A011813 M-sequences from multicomplexes on 4 variables with all monomials of degree 5 but none of degree larger than n.
%Y A011813 Sequence in context: A017801 A017748 A093285 this_sequence A006106 A015338 A131750
%Y A011813 Adjacent sequences: A011810 A011811 A011812 this_sequence A011814 A011815 A011816
%K A011813 nonn
%O A011813 1,7
%A A011813 Svante Linusson (linusson(AT)math.kth.se)
%I A006106 M5360
%S A006106 1,85,5797,376805,24208613,1550842085,99277752549,6354157930725,
%T A006106 406672215935205,26027119554103525,1665737215212030181,106607206793565997285
%N A006106 Gaussian binomial coefficient [ n,3 ] for q=4.
%D A006106 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%D A006106 J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D A006106 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p, 99.
%D A006106 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006106 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A006106 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%p A006106 A006106:=1/(z-1)/(4*z-1)/(64*z-1)/(16*z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
%Y A006106 Sequence in context: A017748 A093285 A011813 this_sequence A015338 A131750 A068749
%Y A006106 Adjacent sequences: A006103 A006104 A006105 this_sequence A006107 A006108 A006109
%K A006106 nonn
%O A006106 3,2
%A A006106 njas
%I A015338
%S A015338 1,85,14535,1652145,225683007,28005209505,3642010817055,
%T A015338 462535373765985,59438516325245343,7593183562134412385,
%U A015338 972884994173649887135,124468028808034701006945
%V A015338 1,-85,14535,-1652145,225683007,-28005209505,3642010817055,
%W A015338 -462535373765985,59438516325245343,-7593183562134412385,
%X A015338 972884994173649887135,-124468028808034701006945
%N A015338 Gaussian binomial coefficient [ n,7 ] for q=-2.
%D A015338 J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
%D A015338 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p, 99.
%D A015338 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%Y A015338 Sequence in context: A093285 A011813 A006106 this_sequence A131750 A068749 A045916
%Y A015338 Adjacent sequences: A015335 A015336 A015337 this_sequence A015339 A015340 A015341
%K A015338 sign,easy
%O A015338 7,2
%A A015338 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A131750
%S A131750 1,85,16381,3177721,616461385,119590330861,23199907725541,
%T A131750 4500662508423985,873105326726527441,169377932722437899461,
%U A131750 32858445842826225967885,6374369115575565399870121
%N A131750 Numbers that are both triangular and centered triangular.
%C A131750 We solve r^2+(r+1)^2=0.5*(3*p^2+3*p+2) equivalent to (4*r+2)^2=3*(2*p+1)^2+1.
%C A131750 The diophantine equation X^2=3*Y^2+1 gives X by A001075 and Y by A013453. The return to r gives the sequence 0,6,90,1260,17556,... which satisfies the formulae a(n+2)=14*a(n+1)-a(n)+6 and a(n+1)=7*a(n)+3+(48*a(n)^2+48*a(n)+9)^0.5 and the return to p the sequence A001921 which verifies this new relation : a(n+1)=7*a(n)+(48*a(n)^2+48*a(n)+16)^0.5. Then we obtain the present sequence
%F A131750 a(n+2)=194*a(n+1)-a(n)-108 a(n+1)=97*a(n)-54+14*(48*a(n)^2-54*a(n)+15)^0.5 G.f.: h(z)=a(1)z+a(2)*z^2+...=((z*(1-110*z+z^2)/((1-z)*(1-194*z+z^2))
%p A131750 A131750 := proc(n) coeftayl(x*(1-110*x+x^2)/(1-x)/(1-194*x+x^2),x=0,n) ; end: seq(A131750(n),n=1..20) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 24 2007
%Y A131750 Cf. A001921 A001075 A001353.
%Y A131750 Sequence in context: A011813 A006106 A015338 this_sequence A068749 A045916 A033406
%Y A131750 Adjacent sequences: A131747 A131748 A131749 this_sequence A131751 A131752 A131753
%K A131750 nonn
%O A131750 1,2
%A A131750 Richard Choulet (richardchoulet(AT)yahoo.fr), Sep 20 2007
%E A131750 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 24 2007
%I A068749
%S A068749 1,85,87825,1133641543,184422574177355,379827509377146575439,
%T A068749 9928923943603018785634645661
%N A068749 Number of potential flows in n X n array with integer velocities in -2..2, i.e. number of n X n arrays with adjacent elements differing by no more than 2, counting arrays differing by a constant only once.
%Y A068749 By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, solenoidal flows A068722-A068738.
%Y A068749 Sequence in context: A006106 A015338 A131750 this_sequence A045916 A033406 A020