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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A282673 A280051 A030121 * A062535 A063667 A171063

Adjacent sequences:  A320656 A320657 A320658 * A320660 A320661 A320662

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 18 2018

STATUS

approved

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Last modified March 26 12:43 EDT 2019. Contains 321497 sequences. (Running on oeis4.)