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A322076
Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.
0
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, 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, 0, 0, 1, 0, 20, 0, 0, 2, 0, 0, 0, 0, 6, 16, 0, 0, 1, 0
OFFSET
1,16
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(90) = 7 set multipartitions of {1,1,1,2,2,3,3,4} with no singletons:
{{1,2},{1,2},{1,3},{3,4}}
{{1,2},{1,3},{1,3},{2,4}}
{{1,2},{1,3},{1,4},{2,3}}
{{1,2},{1,3},{1,2,3,4}}
{{1,2},{1,2,3},{1,3,4}}
{{1,3},{1,2,3},{1,2,4}}
{{1,4},{1,2,3},{1,2,3}}
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
sqnopfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sqnopfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], !PrimeQ[#]&&SquareFreeQ[#]&]}]];
Table[Length[sqnopfacs[Times@@Prime/@nrmptn[n]]], {n, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 25 2018
STATUS
approved