A013955 a(n) = sigma_7(n), the sum of the 7th powers of the divisors of n.

1, 129, 2188, 16513, 78126, 282252, 823544, 2113665, 4785157, 10078254, 19487172, 36130444, 62748518, 106237176, 170939688, 270549121, 410338674, 617285253, 893871740, 1290094638, 1801914272, 2513845188, 3404825448, 4624699020, 6103593751, 8094558822, 10465138360

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

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

References

• Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, p. 51.
• Jean-Pierre Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chap. VII, Section 4., p. 93.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• D. B. Lahiri, Some arithmetical identities for Ramanujan's and divisor functions, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 3 p. 308.
• Don Zagier, Elliptic modular forms and their applications, The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_8(z).
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A001157-A001160, A013666, A013954-A013972, A017665-A017712, A087115.

Programs

Magma

[DivisorSigma(7,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

Mathematica

lst={};Do[AppendTo[lst,DivisorSigma[7,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[7,Range[30]] (* Harvey P. Dale, Dec 10 2016 *)

PARI

a(n)=if(n<1,0,sigma(n,7))

Sage

[sigma(n, 7) for n in range(1, 23)]  # Zerinvary Lajos, Jun 04 2009

Formula

Let sigma(p,n) be the sum of the p-th powers of the divisors of n. Then sigma(7,n) = sigma(3,n) + 120 sum(sigma(3,k) sigma(3,n-k),k=1..n-1) (Cf. A087115). - Eugene Salamin, Apr 29 2006 [Hurwitz Identity, Math. Werke I, 1-66, p. 50, last line. See, e.g., the Koecher-Krieg reference, p. 51, rewritten. - Wolfdieter Lang, Jan 20 2016]

G.f.: Sum_{k>=1} k^7*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^6)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017

From Amiram Eldar, Oct 29 2023: (Start)

Multiplicative with a(p^e) = (p^(7*e+7)-1)/(p^7-1).

Dirichlet g.f.: zeta(s)*zeta(s-7).

Sum_{k=1..n} a(k) = zeta(8) * n^8 / 8 + O(n^9). (End)