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Number of multiset partitions with no constant parts of a multiset whose multiplicities are the prime indices of n.
5

%I #4 Dec 09 2018 12:28:57

%S 1,0,0,1,0,1,0,1,2,1,0,2,0,1,2,4,0,4,0,3,3,1,0,7,4,1,9,4,0,7,0,11,3,1,

%T 5,15,0,1,4,11

%N Number of multiset partitions with no constant parts 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(30) = 7 multiset partitions:

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

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

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

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

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

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

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

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

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t Table[Length[Select[mps[nrmptn[n]],Min@@Length/@Union/@#>1&]],{n,20}]

%Y Cf. A000688, A000961, A001055, A001597, A023893, A023894, A181821, A318284, A320322, A321407, A321760, A322260, A322452.

%K nonn,more

%O 1,9

%A _Gus Wiseman_, Dec 09 2018