A282036 a(n) is the sum of quadratic nonresidues of A002145(n) (the n-th prime == 3 mod 4).

2, 14, 33, 95, 161, 279, 473, 658, 944, 1139, 1491, 1738, 1826, 2884, 2996, 4318, 4585, 5004, 6191, 6683, 7849, 8413, 10314, 10746, 11394, 13157, 13393, 16013, 16566, 18936, 19783, 20376, 23946, 27057, 27804, 30883, 35541, 35232, 36384, 39832, 45671, 50858, 51363, 50059, 55097, 56040

Offset: 1

Author

N. J. A. Sloane, Feb 20 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Crossrefs

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Cf. A002145.

Programs

Maple

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 200 do
p:=ithprime(i1);
if (p mod 4) = 3 then
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
fi;
od:
a; m; d; # A282035, A282036, A282037
# Alternative:
f:= p -> add(-k^2 mod p, k=1..(p-1)/2)::
map(f, select(isprime, [seq(p,p=3..1000,4)])); # Robert Israel, Nov 09 2020

Mathematica

f[p_] := Total[Range[p-1] ~Complement~ Table[Mod[k^2, p], {k, (p-1)/2}] ]; f /@ Select[Range[3, 1000, 4], PrimeQ] (* Jean-François Alcover, Feb 16 2018, after Robert Israel *)

PARI

lista(nn) = forprime(p=2, nn, if(p%4==3, print1(sum(k=1, p-1, if (!issquare(Mod(k, p)), k)), ", "))); \\ Michel Marcus, Nov 09 2020

Formula

a(n) = Sum_{k=1..(A002145(n)-1)/2} (-k^2) mod A002145(n). - J. M. Bergot and Robert Israel, Nov 09 2020