A381951 Nonsquarefree odd numbers whose prime factorization has an odd maximum exponent.

27, 125, 135, 189, 243, 297, 343, 351, 375, 459, 513, 621, 675, 783, 837, 875, 945, 999, 1029, 1107, 1125, 1161, 1215, 1269, 1323, 1331, 1375, 1431, 1485, 1593, 1625, 1647, 1701, 1715, 1755, 1809, 1917, 1971, 2079, 2125, 2133, 2187, 2197, 2241, 2295, 2375, 2403, 2457

Offset: 1

Author

Amiram Eldar, Mar 11 2025

Keywords

• nonn
• easy

Comments

Nonsquarefree odd numbers k such that A051903(k) is odd, or equivalently, odd numbers k such that A051903(k) is an odd number that is larger than 1.

The asymptotic density of this sequence is (1/2) * Sum_{k>=3} (-1)^(k+1) * (1 - 2^k/((2^k-1)*zeta(k))) = 0.019098071299657074975... .

Examples

			125 = 5^3 is a term since it is odd, divisible by a square (25, and thus it is nonsquarefree), and the maximum exponent in its prime factorization is 3, which is odd.
		

Links

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

Crossrefs

Intersection of A005408 and A376142.

Complement of A381950 within A013929.

Cf. A051903.

Programs

Mathematica

q[n_] := n > 1 && OddQ[n]; Select[Range[1, 2500, 2], q[Max[FactorInteger[#][[;; , 2]]]] &]

PARI

isok(k) = k > 1 && k % 2 && apply(x -> (x > 1 && x % 2), vecmax(factor(k)[, 2]));