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Search: id:A060640
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| A060640 |
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If n = Product p_i^e_i then a(n) = Product (1 + 2*p_i + 3*p_i^2 + ... + (e_i+1)*p_i^e_i). |
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16
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| 1, 5, 7, 17, 11, 35, 15, 49, 34, 55, 23, 119, 27, 75, 77, 129, 35, 170, 39, 187, 105, 115, 47, 343, 86, 135, 142, 255, 59, 385, 63, 321, 161, 175, 165, 578, 75, 195, 189, 539, 83, 525, 87, 391, 374, 235, 95, 903, 162, 430, 245, 459, 107, 710, 253, 735, 273, 295, 119
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Equals row sums of triangle A143313 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 06 2008]
Equals row sums of triangle A127099 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2008
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REFERENCES
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D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,1000
Matthew M. Conroy, Home page (listed instead of email address)
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FORMULA
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a(n)=Sum_{d|n} d*tau(d), where tau(d) is the number of divisors of d, cf. A000005. a(n)=Sum_{d|n} d*sigma(n/d), where sigma(n)=sum of divisors of n, cf. A000203. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 23 2001
Multiplicative with a(p^e) = ((e+1)*p^{e+2} - (e+2)*p^{e+1} + 1) / (p-1)^2. Dirichlet g.f.: zeta(s)*zeta(s-1)^2. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 03 2006
(LGF): sum(A060640(n)*x^n/n) = - log( prod(j=1,N, eta(x^(j))) ) - Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
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EXAMPLE
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a(4) = a(2^2) = 1 + (2)(2) + (3)(2^2) = 17; a(6) = a(2)a(3) = (1 + (2)(2))(1+(2)(3)) = (5)(7) = 35.
a(6) = tau(1) + 2*tau(2) + 3*tau(3) + 6*tau(6) = 1 + 2*2 + 3*2 + 6*4 = 35.
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MAPLE
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A060640 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(1+sum((j+1)*ifactors(n)[2][i][1]^j, j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end:
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PROGRAM
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(PARI) j=[]; for(n=1, 200, j=concat(j, sumdiv(n, d, n/d*sigma(d)))); j
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-p*X)^2)[n]) (from R. Stephan)
(PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
N=17; default(seriesprecision, N); x=z+O(z^(N+1))
c=sum(j=1, N, j*x^j);
t=1/prod(j=1, N, eta(x^(j)))
t=log(t)
t=serconvol(t, c)
Vec(t)
(PARI) { for (n=1, 1000, write("b060640.txt", n, " ", direuler(p=2, n, 1/(1 - X)/(1 - p*X)^2)[n]); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 08 2009]
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CROSSREFS
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Cf. A000203, A049060, A057723, A000005, A057660, A001001, A006171.
A143313 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 06 2008]
Cf. A127099.
Sequence in context: A082878 A106506 A029649 this_sequence A064944 A070372 A082818
Adjacent sequences: A060637 A060638 A060639 this_sequence A060641 A060642 A060643
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Apr 17 2001
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Vladeta Jovovic (vladeta(AT)eunet.rs) and Matthew M. Conroy, Apr 17 2001
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