Where records occur in the number of times n = x^2 + y^2 + z^2 + w^2 with x + y + z + w a square and with (x,y,z,w) integers and |x| <= |y| <= |z| <= |w|.
0, 2, 3, 4, 16, 58, 90, 130, 178, 234, 258, 354, 378, 450, 594, 630, 738, 858, 978, 1050, 1098, 1170, 1218, 1314, 1458, 1530, 1638, 1890, 2058, 2070, 2178, 2610, 2754, 2898, 3150, 3570, 3690, 3978, 4158, 4338, 4410, 4914, 5250, 5418, 5670, 5850, 6678
1
Where records occur in S000884.
T. D. Noe, Plot of 82 terms
T. D. Noe, Table of 82 terms
Yu-Chen Sun and Zhi-Wei Sun, Two refinements of Lagrange's four-square theorem, arxiv 1605.03074 (May 10 2016)
(Mma) lim = 10; nn = (lim+1)^2; t = Table[0, {nn}]; Do[n = x^2 + y^2 + z^2 + w^2; If[n < nn && IntegerQ[Sqrt[x + y + z + w]], t[[n+1]]++], {x, -lim, lim}, {y, -Abs[x], Abs[x]}, {z, -Abs[y], Abs[y]}, {w, -Abs[z], Abs[z]}]; mx = -1; tMax = {}; Do[If[k > mx, mx = k; AppendTo[tMax, k]], {k, t}]; Table[Position[t, b, 1, 1][[1, 1]] - 1, {b, tMax}]
nonn
T. D. Noe, May 16 2016