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Search: id:A000037
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| A000037 |
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Numbers that are not squares (note the remarkable formula for the n-th term).
(Formerly M0613 N0223)
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+0
40
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| 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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These are the natural numbers with an even number of divisors. The number of divisors is odd for the complementary sequence, the squares (sequence A000290) and the numbers for which the number of divisors is divisible by 3 is sequence A059269. - Ola Veshta (olaveshta(AT)my-deja.com), Apr 04 2001
Also, a(n) = largest integer m not equal to n such that n = (floor(n^2/m) + m)/2. - Alexander R. Povolotsky (pevnev(AT)juno.com), Feb 10 2008
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REFERENCES
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A. J. dos Reis and D. M. Silberger, "Generating nonpowers by formula", Mathematics Magazine 63 (1990), pp. 53-55.
M. A. Nyblom, "Some curious sequences involving floor and ceiling functions", American Mathematical Monthly 109 (2002), pp. 559-564.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..10000
S. R. Finch, Class number theory
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Continued Fraction
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FORMULA
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a(n) = n + [1/2 + sqrt(n)].
Another formula: a(n) = n + [ sqrt( n + [ sqrt n ] ) ].
a(n) = A000194(n) + n = floor(1/2 *(1 + sqrt(4*n-3)))+ n. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 14 2009]
d(a(n))=even. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 20 2009]
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EXAMPLE
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For example note that the squares 1, 4, 9, 16 are not included.
a(A002061(n)) = a(n^2-n+1) = A002522(n) = n^2 + 1. A002061(n) = central polygonal numbers (n^2-n+1). A002522(n) = numbers of the form n^2 + 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 21 2009]
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MAPLE
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A000037 := n->n+floor(1/2+sqrt(n));
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MATHEMATICA
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f[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[ f[n], {n, 71}] (from Robert G. Wilson v Sep 24 2004)
f[n_]:=Round[Sqrt[n]]; lst={}; Do[AppendTo[lst, n+f[n]], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]
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PROGRAM
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(MAGMA) [n : n in [1..1000] | not IsSquare(n) ];
(MAGMA) at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; print at, n; end if; end for;
(PARI) a(n)=if(n<0, 0, n+(1+sqrtint(4*n))\2)
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CROSSREFS
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Cf. A007412, A000005, A000290, A059269.
Equals A000194(n) + n.
Cf. A134986.
Sequence in context: A072099 A046841 A164514 this_sequence A028761 A028809 A028785
Adjacent sequences: A000034 A000035 A000036 this_sequence A000038 A000039 A000040
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KEYWORD
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easy,nonn,nice,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
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Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 30 2009
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