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%I #12 Dec 18 2018 05:10:22
%S 1,1,1,2,2,3,2,5,5,5,3,9,4,7,9,15,5,18,6,16,14,10,8,31,17,14,40,25,10,
%T 34,12,52,21,19,27,70,15,25,31,59,18,57,22,38,80,33,27,120,46,67,44,
%U 56,32,172,42,100,61,43,38,141,46,55,143,203,64,91,54,80
%N Number of strict multiset partitions of a multiset whose multiplicities are the prime indices of n.
%H Andrew Howroyd, <a href="/A318286/b318286.txt">Table of n, a(n) for n = 1..500</a>
%F a(n) = A045778(A181821(n)).
%F a(prime(n)^k) = A219585(n, k). - _Andrew Howroyd_, Dec 17 2018
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
%t Table[Length[strfacs[Times@@Prime/@nrmptn[n]]],{n,60}]
%o (PARI)
%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
%o count(sig)={my(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=1/prod(i=1, #p, 1 - x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
%o a(n)={if(n==1, 1, count(sig(n)))} \\ _Andrew Howroyd_, Dec 18 2018
%Y Cf. A001055, A007716, A045778, A181821, A305936, A316980, A317776.
%Y Cf. A318283, A318284, A318285, A318287, A318360, A318361.
%K nonn
%O 1,4
%A _Gus Wiseman_, Aug 23 2018
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