A361012 Multiplicative with a(p^e) = sigma(e), where sigma = A000203.

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 7, 1, 3, 1, 3, 1, 1, 1, 4, 3, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 7, 3, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 3, 12, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 3, 3, 1, 1, 1, 7, 7, 1, 1, 3, 1, 1

Offset: 1

Author

Vaclav Kotesovec, Feb 28 2023

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000203, A049419, A145353, A072911, A327837, A327838.

Programs

Mathematica

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

PARI

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

Python

from math import prod
from sympy import divisor_sigma, factorint
def A361012(n): return prod(divisor_sigma(e) for e in factorint(n).values()) # Chai Wah Wu, Feb 28 2023

Formula

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

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma(e) - sigma(e-1)) / p^e) = 2.96008030202494141048182047811089469392843909592516341... = A361013