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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.
5
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, 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, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0
OFFSET
1,16
COMMENTS
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}.
EXAMPLE
The a(48) = 6 factorizations:
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(48) = 6 multiset partitions:
{{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}}
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
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]&]}]];
Table[Length[qepfacs[Times@@Prime/@nrmptn[n]]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 18 2018
STATUS
approved