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A320658
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.
5
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, 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, 0, 4, 1, 0, 0, 0, 6, 2, 0, 0, 1, 0, 17, 1, 0, 7, 2, 0
OFFSET
1,9
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(84) = 7 factorizations into semiprimes:
84 = (4*4*9*35)
84 = (4*4*15*21)
84 = (4*6*6*35)
84 = (4*6*10*21)
84 = (4*6*14*15)
84 = (4*9*10*14)
84 = (6*6*10*14)
The a(84) = 7 multiset partitions into pairs:
{{1,1},{1,1},{2,2},{3,4}}
{{1,1},{1,1},{2,3},{2,4}}
{{1,1},{1,2},{1,2},{3,4}}
{{1,1},{1,2},{1,3},{2,4}}
{{1,1},{1,2},{1,4},{2,3}}
{{1,1},{2,2},{1,3},{1,4}}
{{1,2},{1,2},{1,3},{1,4}}
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
bepfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[bepfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], PrimeOmega[#]==2&]}]];
Table[Length[bepfacs[Times@@Prime/@nrmptn[n]]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 18 2018
STATUS
approved