A381950 Odd numbers whose prime factorization has an even maximum exponent.

1, 9, 25, 45, 49, 63, 75, 81, 99, 117, 121, 147, 153, 169, 171, 175, 207, 225, 245, 261, 275, 279, 289, 315, 325, 333, 361, 363, 369, 387, 405, 423, 425, 441, 475, 477, 495, 507, 525, 529, 531, 539, 549, 567, 575, 585, 603, 605, 625, 637, 639, 657, 693, 711, 725

Offset: 1

Author

Amiram Eldar, Mar 11 2025

Keywords

• nonn
• easy

Comments

Odd numbers k such that A051903(k) is even.

The asymptotic density of this sequence is (1/2) * Sum_{k>=2} (-1)^k * (1 - 2^k/((2^k-1)*zeta(k))) = 0.075617194130991839249... .

Examples

			9 = 3^2 is a term since it is odd and 2 is even.
45 = 3^2 * 5 is a term since it is odd and 2 is even.
125 = 5^3 is not a term since 3 is odd.
		

Links

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

Crossrefs

Intersection of A005408 and A368714.

Subsequence of A381956.

A381823 is a subsequence.

Cf. A051903, A375039, A381951.

Programs

Mathematica

Select[Range[1, 1000, 2], # == 1 || EvenQ[Max[FactorInteger[#][[;; , 2]]]] &]

PARI

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