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

1,2

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).

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)

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 *)

Prog

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

Crossrefs

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

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved