A320658 - OEIS

%I #8 Oct 19 2018 09:47:57

%S 1,0,1,1,0,0,1,0,2,1,0,2,1,0,0,3,0,0,1,0,2,1,0,0,2,0,5,2,1,3,0,0,0,1,

%T 0,6,1,0,2,4,0,0,1,0,0,1,0,9,3,0,0,2,1,0,2,0,2,0,0,0,1,1,6,15,0,3,0,0,

%U 0,4,1,0,0,0,6,2,0,0,1,0,17,1,0,7,2,0

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

%C 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 The a(84) = 7 factorizations into semiprimes:

%e   84 = (4*4*9*35)

%e   84 = (4*4*15*21)

%e   84 = (4*6*6*35)

%e   84 = (4*6*10*21)

%e   84 = (4*6*14*15)

%e   84 = (4*9*10*14)

%e   84 = (6*6*10*14)

%e The a(84) = 7 multiset partitions into pairs:

%e   {{1,1},{1,1},{2,2},{3,4}}

%e   {{1,1},{1,1},{2,3},{2,4}}

%e   {{1,1},{1,2},{1,2},{3,4}}

%e   {{1,1},{1,2},{1,3},{2,4}}

%e   {{1,1},{1,2},{1,4},{2,3}}

%e   {{1,1},{2,2},{1,3},{1,4}}

%e   {{1,2},{1,2},{1,3},{1,4}}

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];

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

%t Table[Length[bepfacs[Times@@Prime/@nrmptn[n]]],{n,100}]

%Y Cf. A001221, A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320659.

%K nonn

%O 1,9

%A _Gus Wiseman_, Oct 18 2018