A067807 Numbers k such that sigma(k)^2 > 2*sigma(k^2).

24, 36, 40, 48, 60, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 156, 160, 168, 176, 180, 192, 200, 204, 208, 210, 216, 224, 228, 240, 252, 264, 270, 276, 280, 288, 300, 312, 320, 324, 336, 348, 352, 360, 372, 378, 384, 392, 396, 400, 408, 416, 420

Offset: 1

Author

Benoit Cloitre, Feb 07 2002

Keywords

• nonn

Comments

For every n>1 sigma(n)^2 > sigma(n^2).

Limit_{n->oo} a(n)/n appears to exist and is near 8.0; e.g., a(124094) = 1000000. - Paul D. Hanna, Sep 22 2011

We also have a(12438441) = 10^8, a(124240921) = 10^9, and a(1242729194) = 10^10. - Giovanni Resta, Jun 15 2018

All the terms are abundant numbers (A005101). - Amiram Eldar, May 03 2025

Examples

			The limit a(n)/n seems to be near 8.0:
        n     a(n)    a(n)/n
  ------- --------  ----------
   124094  1000000  8.05840...
   248310  2000000  8.05444...
   372503  3000000  8.05362...
   496826  4000000  8.05110...
   621163  5000000  8.04941...
   745602  6000000  8.04718...
   870189  7000000  8.04422...
   994799  8000000  8.04182...
  1119336  9000000  8.04048...
  1243884 10000000  8.03933...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203 (sigma), A195735, A065764.

Subsequence of A005101.

Programs

Mathematica

Select[Range[500],DivisorSigma[1,#]^2>2DivisorSigma[1,#^2]&]  (* Harvey P. Dale, Mar 30 2011 *)

PARI

{for(n=1,8000,if(2*sigma(n^2)-sigma(n)^2 < 0,print1(n,",")))} \\ Paul D. Hanna, Sep 22 2011