|
|
A013971
|
|
a(n) = sigma_23(n), the sum of the 23rd powers of the divisors of n.
|
|
6
|
|
|
1, 8388609, 94143178828, 70368752566273, 11920928955078126, 789730317205170252, 27368747340080916344, 590295880727458217985, 8862938119746644274757, 100000011920928963466734, 895430243255237372246532, 6624738056749922960468044, 41753905413413116367045798
(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
|
Multiplicative with a(p^e) = (p^(23*e+23)-1)/(p^23-1).
Dirichlet g.f.: zeta(s)*zeta(s-23).
Sum_{k=1..n} a(k) = zeta(24) * n^24 / 24 + O(n^25). (End)
|
|
MATHEMATICA
|
|
|
PROG
|
(Magma) [DivisorSigma(23, n): n in [1..30]]; // G. C. Greubel, Nov 03 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|