OFFSET
1,9
COMMENTS
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}.
EXAMPLE
The a(30) = 7 multiset partitions:
{{1,1,1,2,2,3}}
{{1,2},{1,1,2,3}}
{{1,3},{1,1,2,2}}
{{2,3},{1,1,1,2}}
{{1,1,2},{1,2,3}}
{{1,1,3},{1,2,2}}
{{1,2},{1,2},{1,3}}
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[mps[nrmptn[n]], Min@@Length/@Union/@#>1&]], {n, 20}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 09 2018
STATUS
approved