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. - comment from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001.

J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chap. VII, Section 4., p. 93.

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

G.f. sum(k>=1, k^7*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2003

lst={}; Do[AppendTo[lst, DivisorSigma[7, n]], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 11 2009]

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

(Other) sage: [sigma(n, 7)for n in xrange(1, 23)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

Sequence in context: A088719 A034681 A017677 this_sequence A036085 A000541 A023876

Adjacent sequences: A013952 A013953 A013954 this_sequence A013956 A013957 A013958

nonn,mult

N. J. A. Sloane (njas(AT)research.att.com).