|
|
A013957
|
|
a(n) = sigma_9(n), the sum of the 9th powers of the divisors of n.
|
|
23
|
|
|
1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173, 1001953638, 2357947692, 5170140388, 10604499374, 20701400904, 38445332184, 68853957121, 118587876498, 198756808749, 322687697780, 513002215782, 794320419872, 1209627165996, 1801152661464
(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
Note that the sequence is not monotonically increasing, with a(4488) > a(4489) being the first of infinitely many examples. - Charles R Greathouse IV, Dec 28 2021
|
|
LINKS
|
|
|
FORMULA
|
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^8)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/264 = Bernoulli(10)/20. - Vaclav Kotesovec, May 07 2023
Multiplicative with a(p^e) = (p^(9*e+9)-1)/(p^9-1).
Dirichlet g.f.: zeta(s)*zeta(s-9).
Sum_{k=1..n} a(k) = zeta(10) * n^10 / 10 + O(n^11). (End)
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) a(n)=if(n<1, 0, sigma(n, 9))
(Magma) [DivisorSigma(9, n): n in [1..20]]; // Bruno Berselli, Apr 10 2013
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,mult,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|