A303710 - OEIS

%I #14 May 04 2018 08:31:38

%S 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,

%T 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,

%U 5,2,1,9,2,2,2,4,1,9,2,3,2,2,2,6,1,3,3

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

%C 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.

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

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

%t 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]}]

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Apr 29 2018