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

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.

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).

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^7). (End)

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))

(SageMath) [sigma(n, 7) for n in range(1, 28)] # Zerinvary Lajos, Jun 04 2009

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

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

Sequence in context: A034681 A351302 A017677 * A294302 A343509 A221969

Adjacent sequences: A013952 A013953 A013954 * A013956 A013957 A013958

nonn,easy,mult

approved