A366988 The number of prime powers of prime numbers (A053810) that divide n.

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

Offset: 1

Author

Amiram Eldar, Oct 31 2023

Keywords

• nonn
• easy

Comments

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

Every nonnegative number appears in the sequence of record values. k >= 1 first occurs at n = 2^prime(k) (A034785).

Links

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

Crossrefs

Cf. A000720, A005117, A034785, A053810, A095691, A366989, A366990, A366993.

Programs

Mathematica

f[p_, e_] := PrimePi[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); sum(i = 1, #f~, primepi(f[i, 2]));}

Formula

Additive with a(p^e) = A000720(e).

a(n) = 0 if and only if n is squarefree (A005117).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} P(p) = 0.67167522222173297323..., where P(s) is the prime zeta function.