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

1,2

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)

Links

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A023697 A038468 A131348 * A043471 A203857 A305184

Adjacent sequences: A303120 A303121 A303122 * A303124 A303125 A303126

Keyword

nonn

Author

Paolo P. Lava, May 04 2018

Status

approved