A366989 The number of prime powers p^q dividing n, where p is prime and q is either 1 or prime (A334393 without the first term 1).

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 4, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2

Offset: 1

Author

Amiram Eldar, Oct 31 2023

Keywords

• nonn
• easy

Comments

First differs from A122810 at n = 48, and from A318322 at n = 64.

Links

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

Crossrefs

Cf. A000720, A001221, A005117, A077761, A334393, A366988, A366991, A366992, A366994.

Cf. A122810, A318322.

Programs

Mathematica

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

PARI

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

Formula

Additive with a(p^e) = A000720(e) + 1.

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

a(n) = A366988(n) + A001221(n).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{p prime} P(p) = 0.67167522222173297323..., and P(s) is the prime zeta function.