A303123 Numbers whose sum of divisors is the square of one of their divisors.

1, 364, 1080, 1782, 8736, 30256, 86800, 90768, 149856, 632400, 828816, 1033560, 2467600, 8182944, 9587160, 10593720, 12239136, 15487600, 16702800, 23194080, 23556960, 25371360, 33330528, 35746920, 35889480, 36036000, 40753440, 44013120, 45890208, 46462800, 49035168

Offset: 1

Author

Paolo P. Lava, May 04 2018

Keywords

• nonn

Comments

Subset of A090777 and A300906.

From Robert Israel, May 10 2018: (Start)

If m and n are coprime members of the sequence, then m*n is in the sequence.

However, it is not clear whether there are such m and n where neither is 1: in particular, are there odd members other than 1?

Any odd member > 1 is a square greater than 10^14. (End)

Examples

			Divisors of 364 are 1, 2, 4, 7, 13, 14, 26, 28, 52, 91, 182, 364 and their sum is 784 = 28^2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..900 (terms < 10^13)

Crossrefs

Cf. A000203, A090777, A300906, A303993, A303994, A303995, A303996.

Programs

Maple

with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]);
for k from 1 to nops(a) do if sigma(n)=a[k]^2 then print(n); break;
fi; od; od; end: P(10^9);
# Alternative:
filter:= proc(n) local s;
  s:= numtheory:-sigma(n);
  issqr(s) and n^2 mod s = 0
end proc:
select(filter, [$1..10^7]); # Robert Israel, May 10 2018

Mathematica

Reap[For[k = 1, k <= 10^7, k++, If[AnyTrue[Divisors[k], DivisorSigma[1, k] == #^2&], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 05 2020 *)

PARI

isok(n) = (s = sigma(n)) && issquare(s) && !(n % sqrtint(s)); \\ Michel Marcus, May 04 2018