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

1,1

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A000203, A001157.

Subsequence of A005101.

A068403 is a subsequence.

Sequence in context: A054775 A273088 A334674 * A376445 A306776 A300794

Adjacent sequences: A067763 A067764 A067765 * A067767 A067768 A067769

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 04 2002

Status

approved