cp's OEIS Frontend

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).

%I A366989 #7 Nov 02 2023 09:17:40
%S A366989 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,
%T A366989 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,
%U A366989 2,3,1,5,1,2,3,3,2,3,1,4,3,2,1,4,2,2,2
%N 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).
%C A366989 First differs from A122810 at n = 48, and from A318322 at n = 64.
%H A366989 Amiram Eldar, <a href="/A366989/b366989.txt">Table of n, a(n) for n = 1..10000</a>
%F A366989 Additive with a(p^e) = A000720(e) + 1.
%F A366989 a(n) = 1 is and only if n is squarefree (A005117) > 1.
%F A366989 a(n) = A366988(n) + A001221(n).
%F A366989 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.
%t A366989 f[p_, e_] := PrimePi[e] + 1; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366989 (PARI) a(n) = {my(f = factor(n)); sum(i = 1, #f~, 1 + primepi(f[i, 2]));}
%Y A366989 Cf. A000720, A001221, A005117, A077761, A334393, A366988, A366991, A366992, A366994.
%Y A366989 Cf. A122810, A318322.
%K A366989 nonn,easy
%O A366989 1,4
%A A366989 _Amiram Eldar_, Oct 31 2023