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A303710 Number of factorizations of numbers that are not perfect powers using only numbers that are not perfect powers. 6
1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 3, 1, 5, 1, 2, 2, 2, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2, 3, 2, 2, 2, 6, 1, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Note that a factorization of a number that is not a perfect power (A007916) is always itself aperiodic, meaning the multiplicities of its factors are relatively prime.

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

The a(19) = 4 factorizations of 24 are (2*2*2*3), (2*2*6), (2*12), (24).

The a(23) = 5 factorizations of 30 are (2*3*5), (2*15), (3*10), (5*6), (30).

MATHEMATICA

radQ[n_] := And[n > 1, GCD@@FactorInteger[n][[All, 2]] === 1]; facsr[n_] := If[n <= 1, {{}}, Join@@Table[Map[Prepend[#, d] &, Select[facsr[n/d], Min@@# >= d &]], {d, Select[Divisors[n], radQ]}]]; Table[Length[facsr[n]], {n, Select[Range[100], radQ]}]

CROSSREFS

Cf. A001055, A001597, A007716, A007916, A052409, A052410, A281116, A303365, A303386, A303707, A303708, A303709.

Sequence in context: A066075 A072347 A318831 * A263280 A136107 A178691

Adjacent sequences:  A303707 A303708 A303709 * A303711 A303712 A303713

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 29 2018

STATUS

approved

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Last modified July 9 14:56 EDT 2020. Contains 335543 sequences. (Running on oeis4.)