A282035 Sum of quadratic residues of (n-th prime == 3 mod 4).

1, 7, 22, 76, 92, 186, 430, 423, 767, 1072, 994, 1343, 1577, 2369, 2675, 3683, 3930, 4587, 5134, 6520, 6012, 7518, 7831, 8955, 10761, 11596, 12258, 12428, 14809, 15517, 16802, 19527, 23025, 21148, 26811, 29148, 28720, 31929, 35247, 33321, 41900, 41807, 44778, 47844, 51856, 52771, 51253, 57466

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT], 2015.

Andrés Ventas, Fórmula para a suma dos residuos cadráticos dos primos p=4k+3, (2025). (in Galician).

Formula

a(n) = (k - (h(-p(n)) - 1) / 2)*(4k + 3), for n>1, p(n)=A002145(n) primes of type 4k+3, h(-p(n))=A002143(n) are the class numbers. - Andrés Ventas, Nov 29 2025

Example

For n=3, p=11, k=2, h(-p)=1, a(3) = 2*11 = 22.
For n=11, p=71, k=17, h(-p)=7, a(11) = 14*71 = 994.

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

Mathematica

Table[Table[Mod[a^2, p], {a, 1, (p-1)/2}]//Total, {p, Select[Prime[Range[100]], Mod[#, 4]==3 &]}] (* Vincenzo Librandi, Feb 21 2017 *)
Table[Total[PowerMod[#, 2, n]&/@Range[n/2]], {n, Select[Prime[Range[100]], Mod[#, 4]==3&]}] (* Harvey P. Dale, Dec 11 2024 *)

Prog

(PARI) do(p)=sum(k=1, p-1, k^2%p)/2
apply(do, select(p->p%4==3, primes(100))) \\ Charles R Greathouse IV, Feb 21 2017

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.

Sequence in context: A085287 A278767 A286186 * A302273 A151822 A302724

Adjacent sequences: A282032 A282033 A282034 * A282036 A282037 A282038

Keyword

nonn

Author

N. J. A. Sloane, Feb 20 2017

Status

approved