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

1,36

Comments

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

Links

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

Example

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}}

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}]

Prog

(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

Crossrefs

The positions of zeros are A026424.

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

Sequence in context: A348454 A348452 A309163 * A359786 A359763 A277017

Adjacent sequences: A320652 A320653 A320654 * A320656 A320657 A320658

Keyword

nonn

Author

Gus Wiseman, Oct 18 2018

Extensions

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

Status

approved