For the primes p in S000752, the sum_{1..(p-1)/2} Legendre(i/p), which is a new maximum.
1, 3, 5, 9, 15, 21, 27, 33, 45, 51, 69, 87, 93, 99, 105, 117, 129, 135, 141, 147, 165, 183, 195, 201, 207, 213, 225, 243, 261, 273, 285, 297, 309, 315, 325, 327, 357, 363, 387, 393, 399, 429, 459, 495, 525, 561, 573, 579, 591, 597, 621, 669, 693, 747, 759
1
Here Legendre(i/p) is the Legendre symbol.
T. D. Noe, Plot of 100 terms
T. D. Noe, Table of 100 terms
Eric W. Weisstein, MathWorld: Legendre Symbol
(Mma) t = {}; mx = 0; n = 0; While[Length[t] < 50, n++; p = Prime[n]; If[! MemberQ[{1, 5}, Mod[p, 8]], s = Plus @@ JacobiSymbol[Range[(p - 1)/2], p]; If[s > mx, mx = s; AppendTo[t, {p, s}]]]]; Transpose[t][[2]]
Cf. S000752, S000754, S000755.
nonn,hard
T. D. Noe, Nov 20 2015