A320655 Number of factorizations of n into semiprimes. Number of multiset partitions of the multiset of prime factors of n, into pairs.

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0

Offset: 1

Author

Gus Wiseman, Oct 18 2018

Keywords

• nonn

Comments

The characteristic function of nonzero terms is A065043. - R. J. Mathar, Jan 18 2021

Examples

			The a(900) = 5 factorizations into semiprimes:
  900 = (4*9*25)
  900 = (4*15*15)
  900 = (6*6*25)
  900 = (6*10*15)
  900 = (9*10*10)
The a(900) = 5 multiset partitions into pairs:
  {{1,1},{2,2},{3,3}}
  {{1,1},{2,3},{2,3}}
  {{1,2},{1,2},{3,3}}
  {{1,2},{1,3},{2,3}}
  {{2,2},{1,3},{1,3}}
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

The positions of zeros are A026424.

Cf. A001055, A001222, A001358, A007716, A007717, A056239, A112798, A318871, A318953, A320462, A320656, A320658, A320659.

Programs

Mathematica

semfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[semfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[semfacs[n]],{n,100}]

PARI

A320655(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A320655(n/d, d))); (s)); \\ Antti Karttunen, Dec 06 2020

Extensions

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2020