cp's OEIS Frontend

A067766 Numbers k such that sigma(k)^2 > 4*sigma_2(k) where sigma_2(k) is the sum of squares over the divisors of k.

%I A067766 #20 Apr 19 2025 06:15:46
%S A067766 24,36,48,60,72,84,90,96,108,120,126,132,140,144,150,156,160,168,180,
%T A067766 192,204,210,216,228,240,252,264,270,276,280,288,300,312,320,324,330,
%U A067766 336,360,378,384,390,396,400,408,420,432,440,450,456,462,468,480,504
%N A067766 Numbers k such that sigma(k)^2 > 4*sigma_2(k) where sigma_2(k) is the sum of squares over the divisors of k.
%C A067766 From _Amiram Eldar_, Apr 19 2025: (Start)
%C A067766 All the terms are abundant numbers: if k is a term then sigma(k) > 2 * sqrt(sigma_2(k)) >= 2 * sqrt(k^2) = 2*k.
%C A067766 All the 3-abundant numbers (A068403) are terms because sigma_2(k) < zeta(2) * k^2, so 2 * sqrt(sigma_2(k))/k < 2*sqrt(zeta(2)) = 2.565... < 3.
%C A067766 The numbers of terms that do not exceed 10^k, for k = 2, 3, ..., are 8, 109, 1110, 10874, 107610, 1085715, 10872432, 108442685, 1084358031, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1084... . (End)
%H A067766 Amiram Eldar, <a href="/A067766/b067766.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%t A067766 Select[Range[600],DivisorSigma[1,#]^2>4*DivisorSigma[2,#]&] (* _Harvey P. Dale_, Dec 27 2015 *)
%o A067766 (PARI) for(n=1,1000,if(sigma(n)^2>4*sumdiv(n,k,k^2),print1(n,",")))
%o A067766 (PARI) isok(k) = {my(f = factor(k)); sigma(f)^2 > 4 * sigma(f, 2);} \\ _Amiram Eldar_, Apr 19 2025
%Y A067766 Cf. A000203, A001157.
%Y A067766 Subsequence of A005101.
%Y A067766 A068403 is a subsequence.
%K A067766 easy,nonn
%O A067766 1,1
%A A067766 _Benoit Cloitre_, Apr 04 2002