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

Author

Gus Wiseman, Oct 18 2018

Keywords

• nonn

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}.

Examples

			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}}
		

Crossrefs

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

Programs

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}]