A377816 - OEIS

%I #11 Nov 09 2024 08:01:04

%S 4,9,12,16,18,20,25,28,44,45,48,49,50,52,60,63,64,68,72,75,76,80,81,

%T 84,90,92,98,99,108,112,116,117,121,124,126,132,140,147,148,150,153,

%U 156,162,164,169,171,172,175,176,188,192,198,200,204,207,208,212,220,228

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

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

%C 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.

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

%C 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... .

%H Amiram Eldar, <a href="/A377816/b377816.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%Y A377818 is a subsequence.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Nov 09 2024