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%I #18 Dec 15 2020 09:08:13
%S 1,2,2,5,2,6,2,10,5,6,2,16,2,6,6,20,2,16,2,16,6,6,2,36,5,6,10,16,2,22,
%T 2,36,6,6,6,46,2,6,6,36,2,22,2,16,16,6,2,76,5,16,6,16,2,36,6,36,6,6,2,
%U 64,2,6,16,65,6,22,2,16,6,22,2,108,2,6,16,16,6
%N Number of factorizations of n into factors (greater than 1) of two kinds.
%C a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H Andrew Howroyd, <a href="/A301830/b301830.txt">Table of n, a(n) for n = 1..10000</a>
%H Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F Dirichlet g.f.: Product_{n > 1} 1/(1 - n^(-s))^2. [corrected by _Ilya Gutkovskiy_, Dec 14 2020]
%F a(p^n) = A000712(n) for prime p. - _Andrew Howroyd_, Nov 18 2018
%e The a(6) = 6 factorizations: (2*3)*(), (3)*(2), (2)*(3), ()*(2*3), (6)*(), ()*(6).
%e The a(12) = 16 factorizations:
%e ()*(2*2*3), (2)*(2*3), (3)*(2*2), (2*2)*(3), (2*3)*(2), (2*2*3)*(),
%e ()*(2*6), (2)*(6), (6)*(2), (2*6)*(), ()*(3*4), (3)*(4), (4)*(3), (3*4)*(),
%e ()*(12), (12)*().
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Sum[Length[facs[d]]*Length[facs[n/d]],{d,Divisors[n]}],{n,100}]
%o (PARI) MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
%o seq(n)={MultEulerT(vector(n, i, 2))} \\ _Andrew Howroyd_, Nov 18 2018
%Y Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301829.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 27 2018
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