A320658 Number of factorizations of A181821(n) into semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into pairs.

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 5, 2, 1, 3, 0, 0, 0, 1, 0, 6, 1, 0, 2, 4, 0, 0, 1, 0, 0, 1, 0, 9, 3, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 0, 1, 1, 6, 15, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 6, 2, 0, 0, 1, 0, 17, 1, 0, 7, 2, 0

Offset

1,9

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..86.

Example

The a(84) = 7 factorizations into semiprimes:
  84 = (4*4*9*35)
  84 = (4*4*15*21)
  84 = (4*6*6*35)
  84 = (4*6*10*21)
  84 = (4*6*14*15)
  84 = (4*9*10*14)
  84 = (6*6*10*14)
The a(84) = 7 multiset partitions into pairs:
  {{1,1},{1,1},{2,2},{3,4}}
  {{1,1},{1,1},{2,3},{2,4}}
  {{1,1},{1,2},{1,2},{3,4}}
  {{1,1},{1,2},{1,3},{2,4}}
  {{1,1},{1,2},{1,4},{2,3}}
  {{1,1},{2,2},{1,3},{1,4}}
  {{1,2},{1,2},{1,3},{1,4}}

Mathematica

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

Crossrefs

Cf. A001221, A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320659.

Sequence in context: A083650 A138514 A286137 * A284966 A143540 A291336

Adjacent sequences: A320655 A320656 A320657 * A320659 A320660 A320661

Keyword

nonn

Author

Gus Wiseman, Oct 18 2018

Status

approved