A383482 Integers k such that sigma(k)/k - 1 is a rational square.

1, 6, 9, 28, 216, 360, 496, 2016, 2401, 8128, 16758, 182520, 884736, 1022112, 1352328, 1571328, 1935360, 2678400, 33550336, 54758400, 101382400, 119533176, 136808280, 163298502, 198288000, 618591192, 691022088, 782481673, 796663296, 1137067008, 1275418369, 1303102080

Offset: 1

Author

Michel Marcus, Apr 28 2025

Keywords

• nonn

Examples

			6 is a term because sigma(6)/6 - 1 = 2 - 1 = 1, a square; like for all perfect numbers.
9 is a term because sigma(9)/9 - 1 = 13/9 - 1 = 4/9, a square.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..40

Crossrefs

Cf. A000203 (sigma), A069070.

Subsequences: A000396 (perfect numbers), A046060 (5-multiperfect numbers), A381321.

Cf. A218404 (for those terms with sigma(x)/x = 13/4).

Programs

Mathematica

q[k_] := And @@ IntegerQ /@ Sqrt[NumeratorDenominator[DivisorSigma[-1, k] - 1]]; Select[Range[2*10^6], q] (* Amiram Eldar, Apr 28 2025 *)

PARI

isok(k) = issquare(sigma(k)/k - 1);

Extensions

a(30)-a(32) from Jinyuan Wang, Apr 28 2025