cp's OEIS Frontend

A384519 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an even power (A384517).

%I A384519 #10 Jun 01 2025 09:58:41
%S A384519 4,9,12,16,18,20,25,28,36,44,45,48,49,50,52,60,63,64,68,75,76,80,81,
%T A384519 84,90,92,98,99,100,112,116,117,121,124,126,132,140,147,148,150,153,
%U A384519 156,162,164,169,171,172,175,176,180,188,192,196,198,204,207,208,212,220
%N A384519 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an even power (A384517).
%C A384519 Subsequence of A240112 and first differs from it at n = 30: A240112(30) = 108 is not a term of this sequence.
%C A384519 Subsequence of A368714 and differs from it by not having the terms 1, 144, 324, 400, 432, ... .
%C A384519 Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is even.
%C A384519 Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an even power (A384517).
%C A384519 The asymptotic density of this sequence is Sum_{k>=1} (d(2*k)-1)/zeta(2) = 0.265530259454558018819..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i).
%H A384519 Amiram Eldar, <a href="/A384519/b384519.txt">Table of n, a(n) for n = 1..10000</a>
%H A384519 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%t A384519 q[n_] := Module[{u = Union[Select[FactorInteger[n][[;; , 2]], # > 1 &]]}, Length[u] == 1 && EvenQ[u[[1]]]]; Select[Range[250], q]
%o A384519 (PARI) isok(k) = {my(e = select(x -> (x > 1), Set(factor(k)[, 2]))); #e == 1 && !(e[1] % 2);}
%Y A384519 Intersection of A335275 and A375142.
%Y A384519 Intersection of A368714 and A375142.
%Y A384519 Equals A375142 \ A384520.
%Y A384519 Subsequence of A013929 and A240112.
%Y A384519 Subsequences: A067259, A384517.
%Y A384519 Cf. A005117, A013661, A057521.
%K A384519 nonn
%O A384519 1,1
%A A384519 _Amiram Eldar_, Jun 01 2025