A322076 Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.

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

Author

Gus Wiseman, Nov 25 2018

Keywords

• nonn

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}.

Examples

			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}}
		

Crossrefs

Cf. A001055, A007716, A049311, A056156, A116540, A181821, A255906, A304382.

Cf. A318284, A318286, A318360, A318361, A318362, A318369.

Programs

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}]