cp's OEIS Frontend

A320659 Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.

%I A320659 #6 Oct 19 2018 09:48:04
%S A320659 1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,3,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,
%T A320659 0,3,0,0,0,1,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,15,0,0,0,0,
%U A320659 0,1,0,0,0,0,1,0,0,0,0,0,6,0,0,1,0,0,0
%N A320659 Number of factorizations of A181821(n) into squarefree semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into strict pairs.
%C A320659 This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%e A320659 The a(48) = 6 factorizations:
%e A320659   4620 = (6*10*77)
%e A320659   4620 = (6*14*55)
%e A320659   4620 = (6*22*35)
%e A320659   4620 = (10*14*33)
%e A320659   4620 = (10*21*22)
%e A320659   4620 = (14*15*22)
%e A320659 The a(48) = 6 multiset partitions:
%e A320659   {{1,2},{1,3},{4,5}}
%e A320659   {{1,2},{1,4},{3,5}}
%e A320659   {{1,2},{1,5},{3,4}}
%e A320659   {{1,3},{1,4},{2,5}}
%e A320659   {{1,3},{2,4},{1,5}}
%e A320659   {{1,4},{2,3},{1,5}}
%t A320659 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A320659 qepfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[qepfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
%t A320659 Table[Length[qepfacs[Times@@Prime/@nrmptn[n]]],{n,100}]
%Y A320659 A001147(n) = a(2^(2n)).
%Y A320659 Cf. A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320658.
%K A320659 nonn
%O A320659 1,16
%A A320659 _Gus Wiseman_, Oct 18 2018