A384519 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an even power (A384517).

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208, 212, 220

Offset: 1

Author

Amiram Eldar, Jun 01 2025

Keywords

• nonn

Comments

Subsequence of A240112 and first differs from it at n = 30: A240112(30) = 108 is not a term of this sequence.

Subsequence of A368714 and differs from it by not having the terms 1, 144, 324, 400, 432, ... .

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

Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an even power (A384517).

The asymptotic density of this sequence is Sum_{k>=1} (d(2*k)-1)/zeta(2) = 0.265530259454558018819..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/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 A335275 and A375142.

Intersection of A368714 and A375142.

Equals A375142 \ A384520.

Subsequence of A013929 and A240112.

Subsequences: A067259, A384517.

Cf. A005117, A013661, A057521.

Programs

Mathematica

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

PARI

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