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.

24, 36, 48, 60, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 156, 160, 168, 180, 192, 204, 210, 216, 228, 240, 252, 264, 270, 276, 280, 288, 300, 312, 320, 324, 330, 336, 360, 378, 384, 390, 396, 400, 408, 420, 432, 440, 450, 456, 462, 468, 480, 504

Offset: 1

Author

Benoit Cloitre, Apr 04 2002

Keywords

• easy
• nonn

Comments

From Amiram Eldar, Apr 19 2025: (Start)

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.

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.

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)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Crossrefs

Cf. A000203, A001157.

Subsequence of A005101.

A068403 is a subsequence.

Programs

Mathematica

Select[Range[600],DivisorSigma[1,#]^2>4*DivisorSigma[2,#]&] (* Harvey P. Dale, Dec 27 2015 *)

PARI

for(n=1,1000,if(sigma(n)^2>4*sumdiv(n,k,k^2),print1(n,",")))

PARI

isok(k) = {my(f = factor(k)); sigma(f)^2 > 4 * sigma(f, 2);} \\ Amiram Eldar, Apr 19 2025