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A328745 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s))^p.

%I A328745 #24 Mar 28 2025 08:54:30
%S A328745 1,2,3,3,5,6,7,4,6,10,11,9,13,14,15,5,17,12,19,15,21,22,23,12,15,26,
%T A328745 10,21,29,30,31,6,33,34,35,18,37,38,39,20,41,42,43,33,30,46,47,15,28,
%U A328745 30,51,39,53,20,55,28,57,58,59,45,61,62,42,7,65,66,67,51,69,70,71,24,73,74,45
%N A328745 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s))^p.
%C A328745 Number of ways to factor n into 2 kinds of 2, 3 kinds of 3, 5 kinds of 5, ... , p kinds of p.
%H A328745 Vaclav Kotesovec, <a href="/A328745/b328745.txt">Table of n, a(n) for n = 1..10000</a>
%H A328745 Vaclav Kotesovec, <a href="/A328745/a328745.jpg">Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000</a>
%F A328745 If n = Product (p_j^k_j) then a(n) = Product (binomial(p_j + k_j - 1, k_j)).
%F A328745 Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.40373... - _Vaclav Kotesovec_, Mar 28 2025
%p A328745 a:= n-> mul(binomial(i[1]+i[2]-1, i[2]), i=ifactors(n)[2]):
%p A328745 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 26 2019
%t A328745 a[n_] := Times @@ (Binomial[#[[1]] + #[[2]] - 1, #[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]
%o A328745 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^p)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 22 2021
%Y A328745 Cf. A008480, A050367, A059481, A135323, A182938.
%K A328745 nonn,mult
%O A328745 1,2
%A A328745 _Ilya Gutkovskiy_, Oct 26 2019