|
|
A281959
|
|
a(n) = sigma_25(n), the sum of the 25th powers of the divisors of n.
|
|
5
|
|
|
1, 33554433, 847288609444, 1125899940397057, 298023223876953126, 28430288877251865252, 1341068619663964900808, 37778932988857102106625, 717897987692699877379693, 10000000298023223910507558, 108347059433883722041830252, 953962194872104906760006308
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
For k > 0, Sum_{n>=1} sigma_(4*k+1)(n) / exp(2*Pi*n) = Bernoulli(4*k+2)/(8*k+4). For k = 0, Sum_{n>=1} sigma(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = Bernoulli(2)/4 - 1/(8*Pi). - Vaclav Kotesovec, May 07 2023
|
|
LINKS
|
|
|
FORMULA
|
G.f.: Sum_{k>=1} k^25*x^k/(1-x^k).
Sum_{n>=1} a(n)/exp(2*Pi*n) = 657931/24 = Bernoulli(26)/52. - Vaclav Kotesovec, May 07 2023
Multiplicative with a(p^e) = (p^(25*e+25)-1)/(p^25-1).
Dirichlet g.f.: zeta(s)*zeta(s-25).
Sum_{k=1..n} a(k) = zeta(26) * n^26 / 26 + O(n^27). (End)
|
|
EXAMPLE
|
For n = 6: The divisors of 6 are 1, 2, 3, 6, so a(6) = sigma_25(6) = 1^25 + 2^25 + 3^25 + 6^25 = 28430288877251865252. - Felix Fröhlich, Feb 03 2017
|
|
PROG
|
(Python)
from sympy import divisor_sigma
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,mult,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|