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A385531 Numbers x such that there exist three integers 0 and t>0 such that sigma(x)^2 = sigma(y)^2 = sigma(z)^2 = x^2 + y^2 + z^2 + t^2.

%I A385531 #35 Jul 09 2025 23:33:33
%S A385531 4,6,28,45,48,60,156,204,208,360,496,1170,2016,2520,2925,3480,4796,
%T A385531 5532,5733,7152,7605,8128,9680,11050,12402,15776,33468,36720,37064,
%U A385531 38408,43584,50960,55216,63708,70364,83772,92280,106700,114840,116288,149400,163800,166617,167580
%N A385531 Numbers x such that there exist three integers 0<x<=y<=z and t>0 such that sigma(x)^2 = sigma(y)^2 = sigma(z)^2 = x^2 + y^2 + z^2 + t^2.
%C A385531 The numbers x, y, z and t form a sigma-quadratic quadruple. See Dimitrov link.
%H A385531 David A. Corneth, <a href="/A385531/b385531.txt">Table of n, a(n) for n = 1..97</a>
%H A385531 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A385531 David A. Corneth, <a href="/A385531/a385531.gp.txt">PARI program</a>
%H A385531 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A385531 (3480, 3672, 4296, 8520) is such a quadruple because sigma(3480)^2 = sigma(3672)^2 = sigma(4296)^2 = 3480^2 + 3672^2 + 4296^2 + 8520^2.
%o A385531 (PARI) isok(x) = my(s=sigma(x), vi=select(t->(t>=x), invsigma(s))); for (i=1, #vi, for (j=1, #vi, for (k=1, #vi, if ((i==1) || (j==1) || (k==1), my(ss = s^2 - vi[i]^2 - vi[j]^2 - vi[k]^2); if (ss && issquare(ss), return(1)););););); \\ _Michel Marcus_, Jul 09 2025
%o A385531 (PARI) \\ See Corneth link
%Y A385531 Cf. A000203, A000414, A385356, A385397.
%K A385531 nonn,hard
%O A385531 1,1
%A A385531 _S. I. Dimitrov_, Jul 02 2025
%E A385531 Some missing terms added by _Michel Marcus_, Jul 09 2025
%E A385531 More terms from _David A. Corneth_, Jul 09 2025