A381956 Numbers k such that k and the maximum exponent in the prime factorization of k have opposite parities.

1, 2, 6, 8, 9, 10, 14, 22, 24, 25, 26, 30, 32, 34, 38, 40, 42, 45, 46, 49, 54, 56, 58, 62, 63, 66, 70, 72, 74, 75, 78, 81, 82, 86, 88, 94, 96, 99, 102, 104, 106, 108, 110, 114, 117, 118, 120, 121, 122, 128, 130, 134, 136, 138, 142, 146, 147, 152, 153, 154, 158, 160

Offset: 1

Author

Amiram Eldar, Mar 11 2025

Keywords

• nonn
• easy

Comments

Numbers k such that A000035(k) != A381955(k).

The asymptotic density of this sequence is 1/2 + Sum_{k>=2} ((-1)^(k+1)/((2^k-1)*zeta(k))) = 0.3753176676637566708086... .

Examples

			2 is a term since 2 mod 2 = 0 and A051903(2) mod 2 = 1 mod 2 = 1 != 0.
4 is not a term since 4 mod 2 = 0 and also A051903(4) mod 2 = 2 mod 2 = 0.
		

Links

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

Crossrefs

Cf. A000035, A051903, A381955.

Subsequences: A039956, A381950.

Programs

Mathematica

q[k_] := If[k == 1, True, Mod[k, 2] != Mod[Max[FactorInteger[k][[;; , 2]]], 2]]; Select[Range[160], q]

PARI

isok(k) = if(k == 1, 1, k % 2 != vecmax(factor(k)[,2]) % 2);