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The number of distinct prime factors of the cubefree numbers.
5

%I #11 Sep 22 2024 03:57:30

%S 0,1,1,1,1,2,1,1,2,1,2,1,2,2,1,2,1,2,2,2,1,1,2,2,1,3,1,2,2,2,2,1,2,2,

%T 1,3,1,2,2,2,1,1,2,2,2,1,2,2,2,1,3,1,2,2,2,3,1,2,2,3,1,1,2,2,2,2,3,1,

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

%N The number of distinct prime factors of the cubefree numbers.

%H Amiram Eldar, <a href="/A376365/b376365.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 1.1.

%F a(n) = A001221(A004709(n)).

%F Sum_{A004709(k) <= x} a(k) = (6/Pi^2) * x * (log(log(x)) + B - C) + O(x/log(x)), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-1)/(p*(p^3-1)) = 0.10770743252352371604... (Das et al., 2024).

%t f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Length[e], Nothing]]; Array[f, 150]

%o (PARI) lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] > 2, is = 0; break)); if(is, print1(#e, ", ")));}

%Y Cf. A001221, A004709, A059956, A072047, A077761, A376361, A376363, A376366.

%K nonn,easy

%O 1,6

%A _Amiram Eldar_, Sep 21 2024