cp's OEIS Frontend

A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

%I A375031 #12 Jul 28 2024 16:39:21
%S A375031 4,9,12,18,20,25,28,36,44,45,49,50,52,60,63,68,72,75,76,84,90,92,98,
%T A375031 99,100,108,116,117,121,124,126,132,140,147,148,150,153,156,164,169,
%U A375031 171,172,175,180,188,196,198,200,204,207,212,220,225,228,234,236,242,244,245
%N A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.
%C A375031 Subsequence of A304365 and differs from it by not having the terms 1, 144, 216, 324, 400, ... .
%C A375031 Subsequence of A038109 and differs from it by not having the terms 144, 324, 400, 576, 720, ... .
%C A375031 Numbers whose largest unitary divisor that is a square (A350388) is a square of squarefree number (A062503) that is larger than 1.
%C A375031 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.
%C A375031 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... .
%H A375031 Amiram Eldar, <a href="/A375031/b375031.txt">Table of n, a(n) for n = 1..10000</a>
%H A375031 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A375031 A375033(a(n)) = 2.
%e A375031 4 = 2^2 is a term because it has the exponent 2 in its prime factorization, and no higher even exponent.
%e A375031 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.
%t A375031 q[n_] := Max[Select[FactorInteger[n][[;; , 2]], EvenQ]] == 2; Select[Range[250], q]
%o A375031 (PARI) is(k) = {my(e = select(x -> !(x % 2), factor(k)[,2])); #e > 0 && vecmax(e) == 2;}
%Y A375031 Subsequence of A013929, A038109 and A304365.
%Y A375031 A062503 \ {1} is a subsequence.
%Y A375031 Cf. A065463, A065466, A268335, A335275, A350388, A375033.
%K A375031 nonn,easy
%O A375031 1,1
%A A375031 _Amiram Eldar_, Jul 28 2024