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proposed

allocated for Gus WisemanNumber of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

1, 1, 2, 2, 3, 4, 5, 4, 7, 5, 7, 9, 12, 7, 11, 11, 16, 11, 19, 16, 21, 15, 29, 11, 12, 26, 30, 15, 31, 38, 22, 21, 47, 26, 29, 52, 45, 36, 57, 26, 64, 19, 30, 52, 77, 52, 36, 57, 98, 21, 67, 38, 74, 97, 66, 105, 47, 42, 36, 109, 118, 98, 92, 109, 52, 171, 30

1,3

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. Factorizations are counted by A001055.

The unsorted prime signature of A055932(n) is given by row n of A124829.

a(n) = A001055(A055932(n)).

The a(1) = 1 through a(11) = 7 factorizations:

{} 2 4 6 8 12 16 18 24 30 32

2*2 2*3 2*4 2*6 2*8 2*9 3*8 5*6 4*8

2*2*2 3*4 4*4 3*6 4*6 2*15 2*16

2*2*3 2*2*4 2*3*3 2*12 3*10 2*2*8

2*2*2*2 2*2*6 2*3*5 2*4*4

2*3*4 2*2*2*4

2*2*2*3 2*2*2*2*2

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

Length@*facs/@First/@GatherBy[Range[1500], If[#==1, {}, Last/@FactorInteger[#]]&]

The sorted-signature version is A050322.

This sequence has range A045782.

Factorizations are A001055.

Cf. A025487, A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

allocated

nonn

Gus Wiseman, Jan 10 2020

approved

editing

allocated for Gus Wiseman

allocated

approved