A362870 a(n) = sigma_29(n), the sum of the 29th powers of the divisors of n.

1, 536870913, 68630377364884, 288230376688582657, 186264514923095703126, 36845653355419807219092, 3219905755813179726837608, 154742505198902911050973185, 4710128697246313465298968573, 100000000186264514923632574038, 1586309297171491574414436704892

Offset: 1

Author

Vaclav Kotesovec, May 07 2023

Keywords

• nonn
• easy
• mult

Comments

In general, 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).

This formula can best be understood as a statement about the divided Bernoulli numbers b(n) = B(n) / n. Then you can say: If v is twice an odd number greater than 1 (i.e., v = 4*n + 2, a term of A016825 that is greater than 2), then b(v) = 2 * Sum_{j>=1} sigma_{v - 1}(j) / exp(2*Pi*j) = A358625(v) / A075180(v - 1). - Peter Luschny, May 08 2023

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Peter Luschny, Is this a new representation of (some) Bernoulli numbers?, Mathematics Stack Exchange.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203 (sigma_1), A001160 (sigma_5), A013957 (sigma_9), A013961 (sigma_13), A013965 (sigma_17), A013969 (sigma_21), A281959 (sigma_25).

Cf. A075180, A358625, A016825, A082771.

Programs

Maple

with(NumberTheory): seq(SumOfDivisors(k, 29), k = 1..20);

Mathematica

DivisorSigma[29, Range[20]]

PARI

for(n=1, 20, print1(direuler( p=2, n, 1 / (1 - X) /(1 - p^29*X))[n], ", "))

Python

from sympy import divisor_sigma
def A362870(n): return divisor_sigma(n,29) # Chai Wah Wu, May 07 2023

Formula

G.f.: Sum_{k>=1} k^29 * x^k / (1-x^k).

Dirichlet g.f.: zeta(s-29)*zeta(s).

Sum_{k=1..n} a(k) ~ zeta(30) * n^30 / 30.

Sum_{n>=1} a(n)/exp(2*Pi*n) = 1723168255201/171864 = Bernoulli(30)/60.

Multiplicative with a(p^e) = (p^(29*e+29)-1)/(p^29-1). - Amiram Eldar, Oct 29 2023