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

1,16

Links

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

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).

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A322583 A351439 A102354 * A370222 A372717 A349615

Adjacent sequences: A380396 A380397 A380398 * A380400 A380401 A380402

Keyword

nonn,easy

Author

Amiram Eldar, Jan 23 2025

Status

approved