|
|
A351316
|
|
Sum of the 10th powers of the square divisors of n.
|
|
3
|
|
|
1, 1, 1, 1048577, 1, 1, 1, 1048577, 3486784402, 1, 1, 1048577, 1, 1, 1, 1099512676353, 1, 3486784402, 1, 1048577, 1, 1, 1, 1048577, 95367431640626, 1, 3486784402, 1048577, 1, 1, 1, 1099512676353, 1, 1, 1, 3656161927895954, 1, 1, 1, 1048577, 1, 1, 1, 1048577, 3486784402, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{d^2|n} (d^2)^10.
Multiplicative with a(p) = (p^(20*(1+floor(e/2))) - 1)/(p^20 - 1). - Amiram Eldar, Feb 07 2022
Dirichlet g.f.: zeta(s) * zeta(2*s-20).
Sum_{k=1..n} a(k) ~ (zeta(21/2)/21) * n^(21/2). (End)
|
|
EXAMPLE
|
a(16) = 1099512676353; a(16) = Sum_{d^2|16} (d^2)^10 = (1^2)^10 + (2^2)^10 + (4^2)^10 = 1099512676353.
|
|
MATHEMATICA
|
f[p_, e_] := (p^(20*(1 + Floor[e/2])) - 1)/(p^20 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
|
|
PROG
|
(PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^20*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022
|
|
CROSSREFS
|
Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), this sequence (k=10).
|
|
KEYWORD
|
nonn,easy,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|