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

%I A361012 #27 Jan 08 2025 02:08:39
%S A361012 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,
%T A361012 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,
%U A361012 1,1,1,12,1,1,3,3,1,1,1,7,7,1,1,3,1,1
%N A361012 Multiplicative with a(p^e) = sigma(e), where sigma = A000203.
%H A361012 Amiram Eldar, <a href="/A361012/b361012.txt">Table of n, a(n) for n = 1..10000</a>
%F A361012 Dirichlet g.f.: Product_{p prime} (1 + Sum_{e>=1} sigma(e) / p^(e*s)).
%F A361012 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
%t A361012 g[p_, e_] := DivisorSigma[1, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
%o A361012 (Python)
%o A361012 from math import prod
%o A361012 from sympy import divisor_sigma, factorint
%o A361012 def A361012(n): return prod(divisor_sigma(e) for e in factorint(n).values()) # _Chai Wah Wu_, Feb 28 2023
%o A361012 (PARI) a(n) = vecprod(apply(sigma, factor(n)[, 2])); \\ _Amiram Eldar_, Jan 07 2025
%Y A361012 Cf. A000203, A049419, A145353, A072911, A327837, A327838.
%K A361012 nonn,mult
%O A361012 1,4
%A A361012 _Vaclav Kotesovec_, Feb 28 2023