A282727 Let p = n-th prime == 3 mod 8; a(n) = (sum of quadratic residues mod p that are < p/2) + (sum of all quadratic residues mod p).

2, 35, 108, 567, 1073, 1386, 2132, 3551, 5330, 6003, 8262, 9968, 13860, 16046, 19625, 24957, 29376, 34155, 37541, 44793, 54758, 61217, 68036, 75215, 77688, 85347, 93366, 98912, 101745, 107531, 119583, 129042, 135548, 145607, 149040, 170478, 193356, 205335, 213521, 230373, 243432, 256851, 280016, 294395

Offset: 1

Author

N. J. A. Sloane, Feb 20 2017

Keywords

• nonn

Comments

This is also the (sum of quadratic nonresidues mod p that are < p/2) + (sum of all quadratic nonresidues mod p) (= A282721 + A282723 = A282724 + A282726).

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 A282721-A282726.

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:
v:= proc(x,r) if x <= r then 2*x else x fi end proc:
f:= proc(p) local q, r, j;
  r:= (p-1)/2;
  add(v(j^2 mod p, r), j=1..r)
end proc:
map(f, select(isprime, [seq(i,i=3..1000,8)])); # Robert Israel, Mar 27 2017

Mathematica

v[x_, r_] := If[x <= r, 2*x, x];
f[p_] := Module[{r}, r = (p-1)/2; Sum[v[PowerMod[j, 2, p], r], {j, 1, r}]];
f /@ Select[Range[3, 1000, 8], PrimeQ] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

Python

from sympy import isprime
def v(x, r):
    return 2*x if x<=r else x
def a(p):
    r=(p - 1)//2
    return sum(v((j**2)%p, r) for j in range(1, r + 1))
print([a(p) for p in range(3, 2001, 8) if isprime(p)]) # Indranil Ghosh, Mar 27 2017 translated from  Robert Israel's Maple program