A303123 - OEIS

%I #33 Jun 05 2020 04:17:23

%S 1,364,1080,1782,8736,30256,86800,90768,149856,632400,828816,1033560,

%T 2467600,8182944,9587160,10593720,12239136,15487600,16702800,23194080,

%U 23556960,25371360,33330528,35746920,35889480,36036000,40753440,44013120,45890208,46462800,49035168

%N Numbers whose sum of divisors is the square of one of their divisors.

%C Subset of A090777 and A300906.

%C From _Robert Israel_, May 10 2018: (Start)

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

%C 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?

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

%H Giovanni Resta, <a href="/A303123/b303123.txt">Table of n, a(n) for n = 1..900</a> (terms < 10^13)

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

%p with(numtheory): P:=proc(q) local a,k,n;

%p for n from 1 to q do a:=sort([op(divisors(n))]);

%p for k from 1 to nops(a) do if sigma(n)=a[k]^2 then print(n); break;

%p fi; od; od; end: P(10^9);

%p # Alternative:

%p filter:= proc(n) local s;

%p   s:= numtheory:-sigma(n);

%p   issqr(s) and n^2 mod s = 0

%p end proc:

%p select(filter, [$1..10^7]); # _Robert Israel_, May 10 2018

%t 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 *)

%o (PARI) isok(n) = (s = sigma(n)) && issquare(s) && !(n % sqrtint(s)); \\ _Michel Marcus_, May 04 2018

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

%K nonn

%O 1,2

%A _Paolo P. Lava_, May 04 2018