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

1,1

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... .

Links

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

Index entries for sequences computed from exponents in factorization of n.

Formula

A375033(a(n)) = 2.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Subsequence of A013929, A038109 and A304365.

A062503 \ {1} is a subsequence.

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

Sequence in context: A336594 A304365 A038109 * A067259 A349931 A060687

Adjacent sequences: A375028 A375029 A375030 * A375032 A375033 A375034

Keyword

nonn,easy

Author

Amiram Eldar, Jul 28 2024

Status

approved