A303710 - OEIS

A303710

Number of factorizations of numbers that are not perfect powers using only numbers that are not perfect powers.

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

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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: A368684 A351034 A318831 * A263280 A136107 A178691

Adjacent sequences: A303707 A303708 A303709 * A303711 A303712 A303713

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 29 2018

STATUS

approved