cp's OEIS Frontend

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

%I A303123 #33 Jun 05 2020 04:17:23
%S A303123 1,364,1080,1782,8736,30256,86800,90768,149856,632400,828816,1033560,
%T A303123 2467600,8182944,9587160,10593720,12239136,15487600,16702800,23194080,
%U A303123 23556960,25371360,33330528,35746920,35889480,36036000,40753440,44013120,45890208,46462800,49035168
%N A303123 Numbers whose sum of divisors is the square of one of their divisors.
%C A303123 Subset of A090777 and A300906.
%C A303123 From _Robert Israel_, May 10 2018: (Start)
%C A303123 If m and n are coprime members of the sequence, then m*n is in the sequence.
%C A303123 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 A303123 Any odd member > 1 is a square greater than 10^14. (End)
%H A303123 Giovanni Resta, <a href="/A303123/b303123.txt">Table of n, a(n) for n = 1..900</a> (terms < 10^13)
%e A303123 Divisors of 364 are 1, 2, 4, 7, 13, 14, 26, 28, 52, 91, 182, 364 and their sum is 784 = 28^2.
%p A303123 with(numtheory): P:=proc(q) local a,k,n;
%p A303123 for n from 1 to q do a:=sort([op(divisors(n))]);
%p A303123 for k from 1 to nops(a) do if sigma(n)=a[k]^2 then print(n); break;
%p A303123 fi; od; od; end: P(10^9);
%p A303123 # Alternative:
%p A303123 filter:= proc(n) local s;
%p A303123   s:= numtheory:-sigma(n);
%p A303123   issqr(s) and n^2 mod s = 0
%p A303123 end proc:
%p A303123 select(filter, [$1..10^7]); # _Robert Israel_, May 10 2018
%t A303123 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 A303123 (PARI) isok(n) = (s = sigma(n)) && issquare(s) && !(n % sqrtint(s)); \\ _Michel Marcus_, May 04 2018
%Y A303123 Cf. A000203, A090777, A300906, A303993, A303994, A303995, A303996.
%K A303123 nonn
%O A303123 1,2
%A A303123 _Paolo P. Lava_, May 04 2018