cp's OEIS Frontend

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

%I A320656 #12 Nov 02 2022 20:22:01
%S A320656 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,
%T A320656 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,
%U A320656 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
%N A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.
%H A320656 Antti Karttunen, <a href="/A320656/b320656.txt">Table of n, a(n) for n = 1..100000</a>
%H A320656 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A320656 a(A002110(n)) = A123023(n). - _Antti Karttunen_, Nov 02 2022
%e A320656 The a(4620) = 6 factorizations into squarefree semiprimes:
%e A320656   4620 = (6*10*77)
%e A320656   4620 = (6*14*55)
%e A320656   4620 = (6*22*35)
%e A320656   4620 = (10*14*33)
%e A320656   4620 = (10*21*22)
%e A320656   4620 = (14*15*22)
%e A320656 The a(4620) = 6 multiset partitions into strict pairs:
%e A320656   {{1,2},{1,3},{4,5}}
%e A320656   {{1,2},{1,4},{3,5}}
%e A320656   {{1,2},{1,5},{3,4}}
%e A320656   {{1,3},{1,4},{2,5}}
%e A320656   {{1,3},{2,4},{1,5}}
%e A320656   {{1,4},{2,3},{1,5}}
%e A320656 The a(69300) = 10 factorizations into squarefree semiprimes:
%e A320656   69300 = (6*6*35*55)
%e A320656   69300 = (6*10*15*77)
%e A320656   69300 = (6*10*21*55)
%e A320656   69300 = (6*10*33*35)
%e A320656   69300 = (6*14*15*55)
%e A320656   69300 = (6*15*22*35)
%e A320656   69300 = (10*10*21*33)
%e A320656   69300 = (10*14*15*33)
%e A320656   69300 = (10*15*21*22)
%e A320656   69300 = (14*15*15*22)
%e A320656 The a(69300) = 10 multiset partitions into strict pairs:
%e A320656   {{1,2},{1,2},{3,4},{3,5}}
%e A320656   {{1,2},{1,3},{2,3},{4,5}}
%e A320656   {{1,2},{1,3},{2,4},{3,5}}
%e A320656   {{1,2},{1,3},{2,5},{3,4}}
%e A320656   {{1,2},{1,4},{2,3},{3,5}}
%e A320656   {{1,2},{2,3},{1,5},{3,4}}
%e A320656   {{1,3},{1,3},{2,4},{2,5}}
%e A320656   {{1,3},{1,4},{2,3},{2,5}}
%e A320656   {{1,3},{2,3},{2,4},{1,5}}
%e A320656   {{1,4},{2,3},{2,3},{1,5}}.
%e A320656 The a(210) = 3 factorizations into squarefree semiprimes: 210 = (6*35) = (10*21) = (14*15). - _Antti Karttunen_, Nov 02 2022
%t A320656 bepfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[bepfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
%t A320656 Table[Length[bepfacs[n]],{n,100}]
%o A320656 (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
%Y A320656 Cf. A001055, A002110, A006881, A079275, A007716, A007717, A045778, A123023, A318871, A318953, A320462, A320655, A320658, A320659.
%K A320656 nonn
%O A320656 1,210
%A A320656 _Gus Wiseman_, Oct 18 2018
%E A320656 Data section extended up to a(120) and the secondary offset added by _Antti Karttunen_, Nov 02 2022