A282721 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are < p/2.

1, 13, 32, 137, 306, 314, 555, 876, 1400, 1416, 1742, 2450, 3099, 3788, 4816, 5430, 6351, 7344, 8393, 9546, 12858, 13373, 15265, 17277, 16311, 18403, 19521, 22344, 21805, 23590, 25495, 26805, 30767, 30863, 31570, 35980, 40678, 43946, 45640, 49124, 50055, 52776, 58418, 66210, 71521, 71665, 83666, 81628

Offset: 1

Author

N. J. A. Sloane, Feb 20 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..4000
• Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Crossrefs

Cf. A282035-A282043 and A282722-A282727.

Programs

Maple

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
  q:=q+j;
  if j
A282721 - A282727
# Alternative
f:= proc(p) local q,r,t,j;
  r:= (p-1)/2; t:= 0;
  for j from 1 to r do
    q:= j^2 mod p;
    if q <= r then t:= t+q fi;
od:
t
end proc:
map(f, select(isprime, [seq(i,i=3..10000,8)])); # Robert Israel, Mar 27 2017

Mathematica

s[p_] := Total[Select[Range[Floor[p/2]], JacobiSymbol[#, p] == 1&]];
s /@ Select[Range[3, 2000, 8], PrimeQ] (* Jean-François Alcover, Nov 17 2017 *)

Python

from sympy import isprime
def a(p):
    r=(p - 1)//2
    t=0
    for j in range(1, r + 1):
        q=(j**2)%p
        if q<=r:t+=q
    return t
print([a(p) for p in range(3, 2001, 8) if isprime(p)]) # Indranil Ghosh, Mar 27 2017, translated from Maple code