A048699 Nonprime numbers whose sum of aliquot divisors (A001065) is a perfect square.

1, 9, 12, 15, 24, 26, 56, 75, 76, 90, 95, 119, 122, 124, 140, 143, 147, 153, 176, 194, 215, 243, 287, 332, 363, 386, 407, 477, 495, 507, 511, 524, 527, 536, 551, 575, 688, 738, 791, 794, 815, 867, 871, 892, 924, 935, 963, 992, 1075, 1083, 1159, 1196, 1199, 1295, 1304

Offset: 1

Author

Enoch Haga

Keywords

• easy
• nonn

Comments

The sum of aliquot divisors of prime numbers is 1.

If a^2 is an odd square for which a^2-1 = p + q with p,q primes, then p*q is a term. If m = 2^k-1 is a Mersenne prime then m*(2^k) (twice an even perfect number) is a term. If b = 2^j is a square and b-7 = 3s is a semiprime then 4s is a term. - Metin Sariyar, Apr 02 2020

Examples

			a(3)=15; aliquot divisors are 1,3,5; sum of aliquot divisors = 9 and 3^2=9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001065, A006532, A020477, A048698, A073040 (includes primes).

Programs

Maple

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

Mathematica

nn=1400;Select[Complement[Range[nn],Prime[Range[PrimePi[nn]]]],IntegerQ[ Sqrt[DivisorSigma[1,#]-#]]&] (* Harvey P. Dale, Apr 25 2011 *)

PARI

isok(k) = !ispseudoprime(k) && issquare(sigma(k) - k); \\ Michel Marcus, May 13 2025