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The number of non-unitary prime divisors of the cubefree numbers.
4

%I #11 Sep 22 2024 03:58:02

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

%T 0,0,0,1,1,0,0,1,1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,1,1,0,0,0,

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

%N The number of non-unitary prime divisors of the cubefree numbers.

%H Amiram Eldar, <a href="/A376366/b376366.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/2409.11275">On the number of prime factors with a given multiplicity over h-free and h-full numbers</a>, arXiv:2409.11275 [math.NT], 2024. Theorem 1.2.

%F a(n) = A056170(A004709(n)).

%F a(n) = A369427(A004709(n)).

%F Sum_{A004709(k) <= x} a(k) = c * x + O(sqrt(x)/log(x)), where c = (1/zeta(3)) * Sum_{p prime} (p*(p-1)/(p^3-1)) = 0.24833233043359932037... (Das et al., 2024).

%t f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Count[e, 2], 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(#select(x -> x == 2, e), ", ")));}

%Y Cf. A002117, A004709, A056170, A088453, A369427, A376362, A376365.

%K nonn,easy

%O 1,31

%A _Amiram Eldar_, Sep 21 2024