A377817 Numbers that have more than one even exponent in their prime factorization.

36, 100, 144, 180, 196, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1008, 1044, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1584, 1600, 1620, 1692, 1700, 1764, 1800

Offset: 1

Author

Amiram Eldar, Nov 09 2024

Keywords

• nonn
• easy

Comments

Subsequence of A072413 and differs from it by not having the terms 216, 1000, 1080, 1512, ... .

Each term can be represented in a unique way as m * k^2, where m is an exponentially odd number (A268335) and k is a composite number that is coprime to m.

Numbers k such that A350388(k) is a square of a composite number (A062312 \ {1}).

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p*(p+1))) * (1 + Sum_{p prime} 1/(p^2+p-1)) = 0.032993560887093165933... .

Links

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

Crossrefs

Complement of the union of A268335 and A377816.

Subsequence of A072413.

Cf. A062312, A162645, A350388.

Programs

Mathematica

Select[Range[1800], Count[FactorInteger[#][[;; , 2]], _?EvenQ] > 1 &]

PARI

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