This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384255 #28 Jul 10 2025 12:14:43
%S A384255 2,21,27,123,175,2133,2187,6093,340917,504309,1594323,1895841,5308415,
%T A384255 23006577,62188641
%N A384255 Integers k such that there exists an integer 0<m<k such that sigma(m)^2 + sigma(k)^2 = 2*(m^2+k^2).
%C A384255 The most interesting question that arises here is whether there exist such pairs for which sigma(m) = sigma(k), which would imply that sigma(m)^2 = sigma(k)^2 = m^2+k^2. None have been found for m < k <= 10^8.
%H A384255 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A384255 (13, 21) is such a pair because sigma(13)^2 + sigma(21)^2 = 14^2 + 32^2 = 2*(13^2+21^2).
%t A384255 f[n_]:=f[n]=DivisorSigma[1,n]^2-2*n^2;lst={};Do[AppendTo[lst,f@k];If[MemberQ[lst,-f@k],Print@k],{k,10000}] (* _Giorgos Kalogeropoulos_, May 29 2025 *)
%o A384255 (PARI) isok(k) = for(m=1, k-1, if (sigma(m)^2 + sigma(k)^2 == 2*(m^2+k^2), return(1))); \\ _Michel Marcus_, May 23 2025
%Y A384255 Cf. A063990, A259180, A383484, A383714
%K A384255 nonn,hard,more
%O A384255 1,1
%A A384255 _S. I. Dimitrov_, May 23 2025
%E A384255 a(9)-a(15) from _Giorgos Kalogeropoulos_, May 29 2025