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A013956 a(n) = sigma_8(n), the sum of the 8th powers of the divisors of n. 12
1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 43053283, 100390882, 214358882, 431733666, 815730722, 1481554114, 2563287812, 4311810305, 6975757442, 11064693731, 16983563042, 25700456418, 37828630724, 55090232674, 78310985282, 110523825058, 152588281251 (list; graph; refs; listen; history; text; internal format)
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
LINKS
FORMULA
G.f.: Sum_{k>=1} k^8*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^7)) = 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^(8*e+8)-1)/(p^8-1).
Dirichlet g.f.: zeta(s)*zeta(s-8).
Sum_{k=1..n} a(k) = zeta(9) * n^9 / 9 + O(n^10). (End)
MATHEMATICA
Table[DivisorSigma[8, n], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
PROG
(Sage) [sigma(n, 8)for n in range(1, 21)] # Zerinvary Lajos, Jun 04 2009
(PARI) a(n)=sigma(n, 8) \\ Charles R Greathouse IV, Apr 28, 2011
(Magma) [DivisorSigma(8, n): n in [1..30]]; // Bruno Berselli, Apr 10 2013
CROSSREFS
Sequence in context: A034682 A351303 A017679 * A294303 A036086 A000542
KEYWORD
nonn,mult,easy
AUTHOR
STATUS
approved

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Last modified March 28 16:12 EDT 2024. Contains 371254 sequences. (Running on oeis4.)