A073040 Numbers n such that sum of proper divisors of n is a square.

1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 17, 19, 23, 24, 26, 29, 31, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 75, 76, 79, 83, 89, 90, 95, 97, 101, 103, 107, 109, 113, 119, 122, 124, 127, 131, 137, 139, 140, 143, 147, 149, 151, 153, 157, 163, 167, 173, 176, 179, 181, 191

Offset: 1

Author

N. J. A. Sloane, Aug 24 2002

Keywords

• nonn

Comments

Old name was: Numbers n such that sum of divisors of n, sigma (n), minus n is a square.

All primes are terms, since for p prime, A001065(p)=1 and 1 is a square. - Michel Marcus, Apr 22 2018

Examples

			a(6) = 9 because the divisors of 9 are 1, 3, 9, and (1+3+9)-9 = 4 = 2^2.
The number 10 is not in the sequence because (1+2+5+10)-10 = 8, which is not a square.
a(7) = 11 because (1+11)-11 = 1, a square.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A001065, A048698 (which excludes primes).

Programs

Maple

with(numtheory); a := []; for n from 1 to 2000 do if issqr(sigma(n)-n) then a := [op(a), n]; fi; od: a;

Mathematica

Select[Range[200], IntegerQ[Sqrt[-# + Plus@@Divisors[#]]] &] (* Alonso del Arte, Dec 08 2010 *)

PARI

isok(n) = issquare(sigma(n) - n); \\ Michel Marcus, Apr 22 2018

Formula

{n: A001065(n) in A000290} - R. J. Mathar, Dec 11 2010

Extensions

Name edited by Altug Alkan, Apr 22 2018