A377816 Numbers that have a single even exponent in their prime factorization.

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 108, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228

Offset: 1

Author

Amiram Eldar, Nov 09 2024

Keywords

• nonn
• easy

Comments

First differs from A162645 at n = 239: A162645(239) = 900 = 2^2 * 3^2 * 5^2 is not a term of this sequence.

Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is an exponentially odd number (A268335) and p is a prime that does not divide m.

Numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p*(p+1))) * Sum_{p prime} 1/(p^2+p-1) = 0.26256423811374124133... .

Links

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

Crossrefs

Cf. A056798, A065463, A162645, A229125 (odd analog), A268335, A350388, A377817.

A377818 is a subsequence.

Programs

Mathematica

Select[Range[250], Count[FactorInteger[#][[;; , 2]], _?EvenQ] == 1 &]

PARI

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); #select(x -> !(x%2), e) == 1);