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

1,1

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

Mathematica

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

Prog

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

Crossrefs

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

A377818 is a subsequence.

Sequence in context: A368714 A374904 A162645 * A351575 A348739 A135572

Adjacent sequences: A377813 A377814 A377815 * A377817 A377818 A377819

Keyword

nonn,easy

Author

Amiram Eldar, Nov 09 2024

Status

approved