A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset

1,210

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

Formula

a(A002110(n)) = A123023(n). - Antti Karttunen, Nov 02 2022

Example

The a(4620) = 6 factorizations into squarefree semiprimes:
  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(4620) = 6 multiset partitions into strict pairs:
  {{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}}
The a(69300) = 10 factorizations into squarefree semiprimes:
  69300 = (6*6*35*55)
  69300 = (6*10*15*77)
  69300 = (6*10*21*55)
  69300 = (6*10*33*35)
  69300 = (6*14*15*55)
  69300 = (6*15*22*35)
  69300 = (10*10*21*33)
  69300 = (10*14*15*33)
  69300 = (10*15*21*22)
  69300 = (14*15*15*22)
The a(69300) = 10 multiset partitions into strict pairs:
  {{1,2},{1,2},{3,4},{3,5}}
  {{1,2},{1,3},{2,3},{4,5}}
  {{1,2},{1,3},{2,4},{3,5}}
  {{1,2},{1,3},{2,5},{3,4}}
  {{1,2},{1,4},{2,3},{3,5}}
  {{1,2},{2,3},{1,5},{3,4}}
  {{1,3},{1,3},{2,4},{2,5}}
  {{1,3},{1,4},{2,3},{2,5}}
  {{1,3},{2,3},{2,4},{1,5}}
  {{1,4},{2,3},{2,3},{1,5}}.
The a(210) = 3 factorizations into squarefree semiprimes: 210 = (6*35) = (10*21) = (14*15). - Antti Karttunen, Nov 02 2022

Mathematica

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

Prog

(PARI) A320656(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&issquarefree(d)&&2==bigomega(d), s += A320656(n/d, d))); (s)); \\ Antti Karttunen, Nov 02 2022

Crossrefs

Cf. A001055, A002110, A006881, A079275, A007716, A007717, A045778, A123023, A318871, A318953, A320462, A320655, A320658, A320659.

Sequence in context: A356170 A353629 A339661 * A354819 A322075 A369255

Adjacent sequences: A320653 A320654 A320655 * A320657 A320658 A320659

Keyword

nonn

Author

Gus Wiseman, Oct 18 2018

Extensions

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Nov 02 2022

Status

approved