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A320656
Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.
62
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
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
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