A380398 The number of unitary divisors of n that are perfect powers (A001597).

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

Offset: 1

Author

Amiram Eldar, Jan 23 2025

Keywords

• nonn
• easy

Comments

First differs from A368978 at n = 32, from A007424 and A369163 at n = 36, from A278908, A307848, A358260 and A365549 at n = 64, and from A323308 at n = 72.

a(n) depends only on the prime signature of n (A118914).

The record values are 2^k, for k = 0, 1, 2, ..., and they are attained at A061742(k).

The sum of unitary divisors of n that are perfect powers is A380400(n).

Examples

			a(4) = 2 since 4 have 2 unitary divisors that are perfect powers, 1 and 4 = 2^2.
a(72) = 3 since 72 have 3 unitary divisors that are perfect powers, 1, 8 = 2^3, and 9 = 3^2.
		

Links

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

Crossrefs

Cf. A001597, A005117, A008683, A061742, A077610, A091050, A118914, A323308, A380399, A380400.

Cf. A007424, A278908, A307848, A323308, A358260, A365549, A368978, A369163.

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) - A380399(n).

a(n) = 1 if and only if n is squarefree (A005117).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - Sum_{k>=2} mu(k)*(zeta(k)/zeta(k+1) - 1) = 1.49341326536904597349..., where mu is the Moebius function (A008683).