A063531 Numbers k such that sigma(k) + 1 is a square.

2, 7, 8, 14, 15, 23, 32, 33, 35, 47, 54, 56, 57, 60, 72, 78, 79, 84, 87, 92, 95, 120, 123, 124, 128, 138, 143, 154, 165, 167, 174, 184, 190, 196, 213, 223, 235, 242, 252, 253, 258, 267, 295, 312, 315, 319, 323, 327, 348, 359, 375, 378, 380, 393, 412, 423, 439

Offset: 1

Author

Labos Elemer, Aug 02 2001

Keywords

• nonn

Comments

Numbers k such that A000203(k) = -1 + m^2 for some m.

Examples

			If k = p(p+2) is a product of twin primes (from A037074), then sigma(k) + 1 = 1 + (p+1)(p+3) = (p+2)^2, square of the larger twin. Other solutions can be either special primes = m^2 - 2 or composites like 120: sigma(120) = 120 + 60 + ... + 1 = 360 = 19^2 - 1. Square number solution is, e.g., 196: sigma(196) = 399 = 20^2 - 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A000203, A028871, A037074, A063530, A088580.

Programs

Mathematica

Select[Range[500],IntegerQ[Sqrt[DivisorSigma[1,#]+1]]&] (* Harvey P. Dale, Jul 02 2021 *)

PARI

{ n=0; for (a=1, 10^9, if (issquare(sigma(a) + 1), write("b063531.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

Extensions

Minor edits from Franklin T. Adams-Watters, Aug 29 2009