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

1,16

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}.

Links

Table of n, a(n) for n=1..85.

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}]

Crossrefs

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

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

Sequence in context: A222519 A128131 A321188 * A115633 A115713 A199571

Adjacent sequences: A322073 A322074 A322075 * A322077 A322078 A322079

Keyword

nonn

Author

Gus Wiseman, Nov 25 2018

Status

approved