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A013955
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a(n) = sigma_7(n), the sum of the 7th powers of the divisors of n.
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24
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1, 129, 2188, 16513, 78126, 282252, 823544, 2113665, 4785157, 10078254, 19487172, 36130444, 62748518, 106237176, 170939688, 270549121, 410338674, 617285253, 893871740, 1290094638, 1801914272, 2513845188
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, p. 51.
J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chap. VII, Section 4., p. 93.
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LINKS
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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)
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A034681 A351302 A017677 * A294302 A343509 A221969
Adjacent sequences: A013952 A013953 A013954 * A013956 A013957 A013958
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KEYWORD
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nonn,mult
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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