login
The number of prime powers of prime numbers (A053810) that divide n.
4

%I #12 Nov 02 2023 10:25:11

%S 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,

%T 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,

%U 0,0,0,3,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0

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

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

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

%H Amiram Eldar, <a href="/A366988/b366988.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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.

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

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

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

%K nonn,easy

%O 1,8

%A _Amiram Eldar_, Oct 31 2023