A057523 Number of ways n-th powerful (1) number can be expressed as b^2*c^3.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Henry Bottomley, Sep 02 2000

Keywords

• nonn
• easy

Comments

Always positive. (Would be zero for any non-powerful number.)

The positive terms of A370256. - Amiram Eldar, Feb 23 2024

Examples

			a(40) = 2 since the 40th powerful number is 576 = 24^2*1^3 = 3^2*4^3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A002117, A370256.

Programs

Mathematica

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10000], # > 0 &] (* Amiram Eldar, Feb 23 2024 *)

PARI

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[, 2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(apply(x -> (x+2)\2 - (x+2)\3, e)), ", ")));} \\ Amiram Eldar, Feb 23 2024

Formula

From Amiram Eldar, Feb 23 2024: (Start)

a(n) = A370256(A001694(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3). (End)