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

1,4

Comments

First differs from A360720 at n = 72.

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

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

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).

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A168677 A345939 A140210 * A360720 A010130 A361063

Adjacent sequences: A380397 A380398 A380399 * A380401 A380402 A380403

Keyword

nonn,easy,new

Author

Amiram Eldar, Jan 23 2025

Status

approved