1,2

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/504. - Simon Plouffe, Mar 01 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.

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).

Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_6(z).

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.

Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1). - David W. Wilson, Aug 01 2001

G.f.: sum(k>=1, k^5*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003

Dirichlet g.f.: zeta(s)*zeta(s-5). - R. J. Mathar, Mar 06 2011

G.f. also (1 - E_6(q))/540, with the g.f. E_6 of A013973. See Hardy p. 166, (10.5.7) with R = E_6. - Wolfdieter Lang, Jan 31 2017

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^4)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017

a(n) = Sum_{1 <= i, j, k, l, m <= n} tau(gcd(i, j, k, l, m, n)) = Sum_{d divides n} tau(d) * J_5(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 22 2024

A001160 := proc(n)

numtheory[sigma][5](n);

end proc:

seq(A001160(n), n=1..30) ; # R. J. Mathar, Jan 31 2017

lst={}; Do[AppendTo[lst, DivisorSigma[5, n]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

DivisorSigma[5, Range[30]] (* Harvey P. Dale, Nov 11 2013 *)

(SageMath) [sigma(n, 5) for n in range(1, 30)] # Zerinvary Lajos, Jun 04 2009

(PARI) a(n)=sigma(n, 5) \\ Charles R Greathouse IV, Apr 28 2011

(Magma) [DivisorSigma(5, n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

Cf. A000005, A000203, A001157, A001158, A001159, A013973, A000584 (Mobius transform), A178448 (Dirichlet inverse), A248076 (partial sums).

Sequence in context: A034679 A351300 A017673 * A294300 A271208 A343499

Adjacent sequences: A001157 A001158 A001159 * A001161 A001162 A001163

nonn,easy,mult

approved