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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 (list; graph; refs; listen; history; text; internal format)
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 * A115713 A115633 A199571

Adjacent sequences:  A322073 A322074 A322075 * A322077 A322078 A322079

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 25 2018

STATUS

approved

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Last modified October 18 08:19 EDT 2021. Contains 348066 sequences. (Running on oeis4.)