A339553 Number of ordered factorizations of n into perfect powers > 1.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 8

Offset: 1

Author

Ilya Gutkovskiy, Dec 08 2020

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Perfect Power

Crossrefs

Cf. A001597, A050363, A052485 (positions of 0's), A075802, A294068.

Programs

Mathematica

a[n_] := If[n == 1, n, Sum[If[d < n, Boole[GCD @@ FactorInteger[n/d][[All, 2]] > 1] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 128}]

Formula

a(1) = 1; a(n) = Sum_{d|n, dA075802(n/d) * a(d).