A361063 Multiplicative with a(p^e) = sigma_2(e), where sigma_2 = A001157.

1, 1, 1, 5, 1, 1, 1, 10, 5, 1, 1, 5, 1, 1, 1, 21, 1, 5, 1, 5, 1, 1, 1, 10, 5, 1, 10, 5, 1, 1, 1, 26, 1, 1, 1, 25, 1, 1, 1, 10, 1, 1, 1, 5, 5, 1, 1, 21, 5, 5, 1, 5, 1, 10, 1, 10, 1, 1, 1, 5, 1, 1, 5, 50, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 5, 5, 1, 1, 1, 21, 21, 1, 1, 5

Offset: 1

Author

Vaclav Kotesovec, Mar 01 2023

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001157, A049419, A361012, A361064.

Cf. A327837, A361013.

Programs

Mathematica

g[p_, e_] := DivisorSigma[2, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> sigma(x, 2), factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025

Python

from math import prod
from sympy import factorint, divisor_sigma
def A361063(n): return prod(divisor_sigma(e,2) for e in factorint(n).values()) # Chai Wah Wu, Mar 01 2023

Formula

Dirichlet g.f.: Product_{primes p} (1 + Sum_{e>=1} sigma_2(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma_2(e) - sigma_2(e-1)) / p^e) = 11.343154585178523783556367128387762286267199879648613456124659589127638983...