A380400 The sum of unitary divisors of n that are perfect powers (A001597).

1, 1, 1, 5, 1, 1, 1, 9, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 9, 26, 1, 28, 5, 1, 1, 1, 33, 1, 1, 1, 50, 1, 1, 1, 9, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 28, 1, 9, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 18, 1, 1, 26, 5, 1, 1, 1, 17, 82

Offset: 1

Author

Amiram Eldar, Jan 23 2025

Keywords

• nonn
• easy

Comments

First differs from A360720 at n = 72.

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

Examples

			a(4) = 5 since 4 have 2 unitary divisors that are perfect powers, 1 and 4 = 2^2, and 1 + 4 = 5.
a(72) = 18 since 72 have 3 unitary divisors that are perfect powers, 1, 8 = 2^3, and 9 = 3^2, and 1 + 8 + 9 = 18.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k)/n^(3/2) and Sum_{k=1..n} A360720(k)/n^(3/2), for n = 1..1000000

Crossrefs

Cf. A001597, A005117, A077610, A358347, A360720, A380398.

Programs

Mathematica

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

PARI

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

Formula

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

a(n) <= A360720(n).

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