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

1,1

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.

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<p/2 then ql:=ql+j; else qu:=qu+j; fi;
else
n:=n+j;
if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;
fi;
od;
Ql:=[op(Ql), ql];
Qu:=[op(Qu), qu];
Q:=[op(Q), q];
Nl:=[op(Nl), nl];
Nu:=[op(Nu), nu];
N:=[op(N), n];
Th:=[op(Th), q+ql];
fi;
od:
Ql; Qu; Q; Nl; Nu; N; Th; # 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 *)

Prog

(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

Crossrefs

Cf. A282035-A282043 and A282721-A282726.

Sequence in context: A042459 A276713 A257601 * A042353 A193576 A297538

Adjacent sequences: A282724 A282725 A282726 * A282728 A282729 A282730

Keyword

nonn

Author

N. J. A. Sloane, Feb 20 2017

Status

approved