A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244, 245

Offset: 1

Author

Amiram Eldar, Jul 28 2024

Keywords

• nonn
• easy

Comments

Subsequence of A304365 and differs from it by not having the terms 1, 144, 216, 324, 400, ... .

Subsequence of A038109 and differs from it by not having the terms 144, 324, 400, 576, 720, ... .

Numbers whose largest unitary divisor that is a square (A350388) is a square of squarefree number (A062503) that is larger than 1.

Each term is a product of two coprime numbers: an exponentially odd number (A268335) and a square of a squarefree number (A062503) that is larger than 1.

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = A065466 - A065463 = 0.2432910611445097832029... .

Examples

			4 = 2^2 is a term because it has the exponent 2 in its prime factorization, and no higher even exponent.
144 = 2^4 * 3^2 is not a term because it has the exponent 4 in its prime factorization which is even and larger than 2.
		

Links

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

Crossrefs

Subsequence of A013929, A038109 and A304365.

A062503 \ {1} is a subsequence.

Cf. A065463, A065466, A268335, A335275, A350388, A375033.

Programs

Mathematica

q[n_] := Max[Select[FactorInteger[n][[;; , 2]], EvenQ]] == 2; Select[Range[250], q]

PARI

is(k) = {my(e = select(x -> !(x % 2), factor(k)[,2])); #e > 0 && vecmax(e) == 2;}

Formula

A375033(a(n)) = 2.