This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384422 #7 May 28 2025 10:51:55
%S A384422 0,1,1,1,1,2,1,2,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,3,1,2,2,2,1,3,1,4,2,2,
%T A384422 2,2,1,2,2,3,1,3,1,2,2,2,1,3,1,2,2,2,1,3,2,3,2,2,1,3,1,2,2,2,2,3,1,2,
%U A384422 2,3,1,3,1,2,2,2,2,3,1,3,2,2,1,3,2,2,2
%N A384422 The number of prime powers (not including 1) p^e that divide n such that e is coprime to the p-adic valuation of n.
%H A384422 Amiram Eldar, <a href="/A384422/b384422.txt">Table of n, a(n) for n = 1..10000</a>
%F A384422 Additive with a(p^e) = phi(e), where phi is the Euler totient function (A000010).
%F A384422 Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{p prime} f(1/p) = 0.24136815875213146317..., and f(x) = -x + (1-x)*x*Sum_{k>=1} mu(k)*x^(k-1)/(1-x^k)^2.
%t A384422 f[p_, e_] := EulerPhi[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384422 (PARI) a(n) = vecsum(apply(eulerphi, factor(n)[, 2]));
%Y A384422 Cf. A000010, A072911, A077761, A085548.
%K A384422 nonn,easy
%O A384422 1,6
%A A384422 _Amiram Eldar_, May 28 2025