A331049 - OEIS

A331049

Number 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

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OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..67.

FORMULA

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

EXAMPLE

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

MATHEMATICA

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[#]]&]

CROSSREFS

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.

Sequence in context: A022820 A292259 A241397 * A350127 A320159 A342879

Adjacent sequences: A331046 A331047 A331048 * A331050 A331051 A331052

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 10 2020

STATUS

approved