A380399 The number of nonunitary divisors of n that are perfect powers (A001597).

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Author

Amiram Eldar, Jan 23 2025

Keywords

• nonn
• easy

Examples

			a(16) = 2 since 16 have 2 nonunitary divisors that are perfect powers, 4 = 2^2 and 8 = 2^3.
a(32) = 3 since 32 have 3 nonunitary divisors that are perfect powers, 4 = 2^3, 8 = 2^3, and 16 = 2^4.
		

Links

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

Crossrefs

Cf. A001597, A008683, A048105, A072102, A091050, A380398.

Programs

Mathematica

ppQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := DivisorSum[n, 1 &, !CoprimeQ[#, n/#] && ppQ[#] &]; Array[a, 100]

PARI

a(n) = sumdiv(n, d, gcd(d, n/d) > 1 && (d == 1 || ispower(d)));

Formula

a(n) = Sum_{d|n, gcd(d, n/d) > 1} [d in A001597], where [] is the Iverson bracket.

a(n) = A091050(n) - A380398(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A072102 + Sum_{k>=2} mu(k)*(zeta(k)/zeta(k+1) - 1) = Sum_{k>=2} mu(k)*zeta(k)*(1/zeta(k+1)-1) = 0.38105110303589889319..., where mu is the Moebius function (A008683).