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%I #5 Nov 26 2018 17:04:45
%S 1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,4,0,1,0,0,0,0,0,3,1,0,2,0,0,1,0,11,0,0,
%T 0,5,0,0,0,1,0,0,0,0,1,0,0,13,1,1,0,0,0,7,0,0,0,0,0,3,0,0,1,41,0,0,0,
%U 0,0,1,0,20,0,0,2,0,0,0,0,6,16,0,0,1,0
%N Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.
%C This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%e The a(90) = 7 set multipartitions of {1,1,1,2,2,3,3,4} with no singletons:
%e {{1,2},{1,2},{1,3},{3,4}}
%e {{1,2},{1,3},{1,3},{2,4}}
%e {{1,2},{1,3},{1,4},{2,3}}
%e {{1,2},{1,3},{1,2,3,4}}
%e {{1,2},{1,2,3},{1,3,4}}
%e {{1,3},{1,2,3},{1,2,4}}
%e {{1,4},{1,2,3},{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 sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]];
%t Table[Length[sqnopfacs[Times@@Prime/@nrmptn[n]]],{n,30}]
%Y Cf. A001055, A007716, A049311, A056156, A116540, A181821, A255906, A304382.
%Y Cf. A318284, A318286, A318360, A318361, A318362, A318369.
%K nonn
%O 1,16
%A _Gus Wiseman_, Nov 25 2018
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