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

1,4

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

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A122810, A318322.

Sequence in context: A073811 A125030 A116479 * A318322 A122810 A179953

Adjacent sequences: A366986 A366987 A366988 * A366990 A366991 A366992

Keyword

nonn,easy

Author

Amiram Eldar, Oct 31 2023

Status

approved