Records 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|.
1, 2, 3, 7, 11, 13, 19, 23, 27, 32, 33, 36, 51, 53, 56, 58, 69, 73, 75, 79, 88, 91, 92, 101, 112, 120, 125, 129, 135, 140, 155, 162, 170, 185, 195, 205, 210, 216, 225, 238, 241, 259, 268, 284, 291, 310, 312, 385, 390, 400, 407, 428, 433, 505, 507, 540, 543
1
These are the records 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}]; tMax
nonn
T. D. Noe, May 16 2016