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

4, 12, 18, 20, 28, 36, 44, 50, 52, 60, 64, 68, 76, 84, 90, 92, 98, 100, 116, 124, 126, 132, 140, 148, 150, 156, 162, 164, 172, 180, 188, 196, 198, 204, 212, 220, 228, 234, 236, 242, 244, 252, 260, 268, 276, 284, 292, 294, 300, 306, 308, 316, 320, 332, 338, 340

Offset: 1

Author

Amiram Eldar, Mar 11 2025

Keywords

• nonn
• easy

Comments

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}.

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).

Examples

			4 is a term since A381952(4) = gcd(4, A051903(4)) = gcd(4, 2) = 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A051903, A381952.

Subsequence of A368715 (numbers k such that A381952(k) >= 2).

Programs

Mathematica

q[k_] := GCD[k, Max[FactorInteger[k][[;;, 2]]]] == 2; Select[2*Range[200], q]

PARI

isok(k) = !(k % 2) && gcd(k, vecmax(factor(k)[, 2])) == 2;