A320659 Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0

Offset

1,16

Comments

This multiset 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..87.

Example

The a(48) = 6 factorizations:
  4620 = (6*10*77)
  4620 = (6*14*55)
  4620 = (6*22*35)
  4620 = (10*14*33)
  4620 = (10*21*22)
  4620 = (14*15*22)
The a(48) = 6 multiset partitions:
  {{1,2},{1,3},{4,5}}
  {{1,2},{1,4},{3,5}}
  {{1,2},{1,5},{3,4}}
  {{1,3},{1,4},{2,5}}
  {{1,3},{2,4},{1,5}}
  {{1,4},{2,3},{1,5}}

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
qepfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[qepfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], And[SquareFreeQ[#], PrimeOmega[#]==2]&]}]];
Table[Length[qepfacs[Times@@Prime/@nrmptn[n]]], {n, 100}]

Crossrefs

A001147(n) = a(2^(2n)).

Cf. A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320658.

Sequence in context: A373695 A280051 A030121 * A062535 A063667 A171063

Adjacent sequences: A320656 A320657 A320658 * A320660 A320661 A320662

Keyword

nonn

Author

Gus Wiseman, Oct 18 2018

Status

approved