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A376361 The number of distinct prime factors of the powerful numbers.

%I A376361 #13 Mar 25 2025 01:26:05
%S A376361 0,1,1,1,1,1,1,1,2,1,1,2,1,2,2,1,1,1,2,1,2,2,2,2,1,1,2,1,2,1,1,2,2,2,
%T A376361 2,2,2,1,1,2,1,2,2,2,1,2,2,1,2,3,1,2,2,2,1,2,2,2,2,2,2,2,1,2,1,2,2,2,
%U A376361 2,2,1,2,3,3,1,2,2,2,2,1,2,1,1,1,2,2,1,2,2,2,3,2,2,1,2,2,2,2,1,2,2,2,2,2,2
%N A376361 The number of distinct prime factors of the powerful numbers.
%H A376361 Amiram Eldar, <a href="/A376361/b376361.txt">Table of n, a(n) for n = 1..10000</a>
%H A376361 Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, <a href="https://arxiv.org/abs/2409.10430">Distribution of omega(n) over h-free and h-full numbers</a>, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.2.
%H A376361 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A376361 a(n) = A001221(A001694(n)).
%F A376361 Sum_{A001694(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(2) + L(2, 3) - L(2, 4)) + O(sqrt(x)/log(x)), where c = zeta(3/2)/zeta(3) (A090699), B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), L(2, 3) = 1.07848461669337535407..., and L(2, 4) = 0.57937575954505652569... (Das et al., 2024).
%t A376361 f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 1 &], Length[e], Nothing]]; Array[f, 3500]
%o A376361 (PARI) lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] == 1, is = 0; break)); if(is, print1(#e, ", ")));}
%Y A376361 Cf. A001221, A001694, A072047, A077761, A090699, A376362, A376363, A376365.
%K A376361 nonn,easy
%O A376361 1,9
%A A376361 _Amiram Eldar_, Sep 21 2024