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A376362 The number of unitary divisors that are squares of primes applied to the powerful numbers.

%I A376362 #13 Mar 25 2025 20:08:38
%S A376362 0,1,0,1,0,1,0,0,2,1,0,1,0,2,1,1,0,0,1,1,2,1,0,2,0,0,1,1,1,0,1,1,1,0,
%T A376362 2,2,1,0,1,1,0,0,1,2,0,1,1,1,0,3,1,1,1,0,0,2,1,1,2,2,0,1,0,1,1,1,2,2,
%U A376362 1,1,1,0,3,2,1,1,0,0,1,0,2,0,0,1,1,1,0,1,0,2,2,1,0,1,1,1,2,1,0,1,1,2,1,2,0
%N A376362 The number of unitary divisors that are squares of primes applied to the powerful numbers.
%H A376362 Amiram Eldar, <a href="/A376362/b376362.txt">Table of n, a(n) for n = 1..10000</a>
%H A376362 Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, <a href="https://doi.org/10.1016/j.jnt.2024.08.007">On the number of prime factors with a given multiplicity over h-free and h-full numbers</a>, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; <a href="https://arxiv.org/abs/2409.11275">arXiv preprint</a>, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.3.
%H A376362 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A376362 a(n) = A369427(A001694(n)).
%F A376362 Sum_{A001694(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(2) - 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)), and L(2, 4) = 0.57937575954505652569... (Das et al., 2025).
%t A376362 f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 1 &], Count[e, 2], Nothing]]; Array[f, 3500]
%o A376362 (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(#select(x -> x == 2, e), ", ")));}
%Y A376362 Cf. A001694, A077761, A090699, A369427, A376361, A376364, A376366.
%K A376362 nonn,easy
%O A376362 1,9
%A A376362 _Amiram Eldar_, Sep 21 2024