A384520 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an odd power (A384518).

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536

Offset: 1

Author

Amiram Eldar, Jun 01 2025

Keywords

• nonn

Comments

Subsequence of A301517 and A374459 and first differs from them at n = 85: A374459(85) = A374459(85) = 864 = 2^5 * 3^3 is not a term of this sequence.

First differs from its subsequence A381312 at n = 21: a(21) = 216 = 2^3 * 3^3 is not a term of A381312.

Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is odd.

Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an odd power (A384518).

The asymptotic density of this sequence is Sum_{k>=1} (d(2*k+1)-1)/zeta(2) = 0.095609588748823080455..., where d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Intersection of A268335 and A375142.

Intersection of A295661 and A375142.

Intersection of A376142 and A375142.

Equals A375142 \ A384519.

Subsequence of A301517 and A374459.

Subsequences: A381312, A384518.

Cf. A005117, A013661, A057521.

Programs

Mathematica

q[n_] := Module[{u = Union[Select[FactorInteger[n][[;; , 2]], # > 1 &]]}, Length[u] == 1 && OddQ[u[[1]]]]; Select[Range[250], q]

PARI

isok(k) = {my(e = select(x -> (x > 1), Set(factor(k)[, 2]))); #e == 1 && e[1] % 2;}