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

1,1

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

Links

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

Example

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...

Mathematica

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

Prog

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

Crossrefs

Cf. A000203 (sigma), A195735, A065764.

Subsequence of A005101.

Sequence in context: A376271 A378878 A091192 * A224907 A292352 A307342

Adjacent sequences: A067804 A067805 A067806 * A067808 A067809 A067810

Keyword

nonn

Author

Benoit Cloitre, Feb 07 2002

Status

approved