A372503 The number of prime powers that are noninfinitary divisors of n.

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

Offset: 1

Author

Amiram Eldar, May 04 2024

Keywords

• nonn
• easy

Comments

First differs from A318499 at n = 32.

Links

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

Crossrefs

Cf. A000120, A001222, A036537, A048967, A162643, A318499, A349258.

Programs

Mathematica

f[p_, e_] :=  e + 1 - 2^DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecsum(apply(x -> x + 1 - 1 << hammingweight(x), factor(n)[, 2]));

Formula

Additive with a(p^e) = e - 2^A000120(e) + 1 = A048967(e).

a(n) = A001222(n) - A349258(n).

a(n) = 0 if and only if n is in A036537.

a(n) > 0 if and only if n is in A162643.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.4917971717413486467..., where f(x) = 1/(1-x) - (1-x) * Product_{k>=0} (1 + 2*x^(2^k)).