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%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
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