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A381953 Numbers k such that A381952(k) = 2.

%I A381953 #9 Mar 12 2025 08:24:30
%S A381953 4,12,18,20,28,36,44,50,52,60,64,68,76,84,90,92,98,100,116,124,126,
%T A381953 132,140,148,150,156,162,164,172,180,188,196,198,204,212,220,228,234,
%U A381953 236,242,244,252,260,268,276,284,292,294,300,306,308,316,320,332,338,340
%N A381953 Numbers k such that A381952(k) = 2.
%C A381953 Disjoint union of {2 * m | m is an odd number such that A051903(m) > 0 and is divisible by 4}, {2^e * m | m is an odd number such that A051903(m) == 2 (mod 4), gcd(A051903(m), m) = 1), 1 <= e <= A051903(m)}, and {2^e * m | A051903(m) < e, e == 2 (mod 4), gcd(e, m) = 1}.
%C A381953 The asymptotic density of this sequence is Sum_{k>=1} (1-2^(4*k)/((2^(4*k)-1)*zeta(4*k)) - (1-2^(4*k+1)/((2^(4*k+1)-1)*zeta(4*k+1)))))/4 + Sum_{k>=1} (1-1/2^(4*k-2)) * (1-1/2)/(1-1/2^(4*k-1)) * f(2*k-1,4*k-1)/zeta(4*k-1) - (1-1/2^(4*k-3)) * f(4*k-2,4*k-2)/zeta(4*k-2) = 0.16205634516436945215..., where f(n,e) = Product_{prime p|n} (1-1/p)/(1-1/p^e).
%H A381953 Amiram Eldar, <a href="/A381953/b381953.txt">Table of n, a(n) for n = 1..10000</a>
%e A381953 4 is a term since A381952(4) = gcd(4, A051903(4)) = gcd(4, 2) = 2.
%t A381953 q[k_] := GCD[k, Max[FactorInteger[k][[;;, 2]]]] == 2; Select[2*Range[200], q]
%o A381953 (PARI) isok(k) = !(k % 2) && gcd(k, vecmax(factor(k)[, 2])) == 2;
%Y A381953 Cf. A051903, A381952.
%Y A381953 Subsequence of A368715 (numbers k such that A381952(k) >= 2).
%K A381953 nonn,easy
%O A381953 1,1
%A A381953 _Amiram Eldar_, Mar 11 2025