%I #9 Dec 10 2018 16:44:02

%S 1,1,1,2,1,2,1,5,3,2,1,6,1,2,3,15,1,9,1,6,3,2,1,21,4,2,16,6,1,10,1,52,

%T 3,2,4,35,1,2,3,22,1,10,1,6,19,2,1,83,5,13,3,6,1,66,4,22,3,2,1,41,1,2,

%U 20,203,4,10,1,6,3,14,1,153,1,2,26,6,5,10,1

%N Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

%H Andrew Howroyd, <a href="/A318360/b318360.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A050320(A181821(n)).

%F From _Andrew Howroyd_, Dec 10 2018:(Start)

%F a(p) = 1 for prime(p).

%F a(prime(i)*prime(j)) = min(i,j) + 1.

%F a(prime(n)^k) = A188392(n,k). (End)

%e The a(12) = 6 set multipartitions of {1,1,2,3}:

%e {{1},{1,2,3}}

%e {{1,2},{1,3}}

%e {{1},{1},{2,3}}

%e {{1},{2},{1,3}}

%e {{1},{3},{1,2}}

%e {{1},{1},{2},{3}}

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];

%t Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,80}]

%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(n=vecsum(sig), s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + O(x*x^n))); s+=prod(i=1, #sig, polcoef(q,sig[i]))*permcount(p)); s/n!}

%o a(n)={if(n==1, 1, my(s=sig(n)); if(#s<=2, if(#s==1, 1, min(s[1],s[2])+1), count(sig(n))))} \\ _Andrew Howroyd_, Dec 10 2018

%Y Cf. A001055, A007716, A049311, A116540, A181821, A188392, A255906.

%Y Cf. A318283, A318284, A318286, A318361, A318362, A318369.

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 24 2018