cp's OEIS Frontend

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

%I A282721 #32 Mar 24 2021 09:57:46
%S A282721 1,13,32,137,306,314,555,876,1400,1416,1742,2450,3099,3788,4816,5430,
%T A282721 6351,7344,8393,9546,12858,13373,15265,17277,16311,18403,19521,22344,
%U A282721 21805,23590,25495,26805,30767,30863,31570,35980,40678,43946,45640,49124,50055,52776,58418,66210,71521,71665,83666,81628
%N A282721 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are < p/2.
%H A282721 Robert Israel, <a href="/A282721/b282721.txt">Table of n, a(n) for n = 1..4000</a>
%H A282721 Christian Aebi and Grant Cairns, <a href="http://arxiv.org/abs/1512.00896">Sums of Quadratic residues and nonresidues</a>, arXiv:1512.00896 [math.NT], 2015.
%p A282721 with(numtheory):
%p A282721 Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
%p A282721 for i1 from 1 to 300 do
%p A282721 p:=ithprime(i1);
%p A282721 if (p mod 8) = 3 then
%p A282721 ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
%p A282721 for j from 1 to p-1 do
%p A282721 if legendre(j,p)=1 then
%p A282721   q:=q+j;
%p A282721   if j<p/2 then ql:=ql+j; else qu:=qu+j; fi;
%p A282721 else
%p A282721   n:=n+j;
%p A282721   if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;
%p A282721 fi;
%p A282721                     od;
%p A282721 Ql:=[op(Ql),ql];
%p A282721 Qu:=[op(Qu),qu];
%p A282721 Q:=[op(Q),q];
%p A282721 Nl:=[op(Nl),nl];
%p A282721 Nu:=[op(Nu),nu];
%p A282721 N:=[op(N),n];
%p A282721 Th:=[op(Th),q+ql];
%p A282721 fi;
%p A282721 od:
%p A282721 Ql; Qu; Q; Nl; Nu; N; Th; # A282721 - A282727
%p A282721 # Alternative
%p A282721 f:= proc(p) local q,r,t,j;
%p A282721   r:= (p-1)/2; t:= 0;
%p A282721   for j from 1 to r do
%p A282721     q:= j^2 mod p;
%p A282721     if q <= r then t:= t+q fi;
%p A282721 od:
%p A282721 t
%p A282721 end proc:
%p A282721 map(f, select(isprime, [seq(i,i=3..10000,8)])); # _Robert Israel_, Mar 27 2017
%t A282721 s[p_] := Total[Select[Range[Floor[p/2]], JacobiSymbol[#, p] == 1&]];
%t A282721 s /@ Select[Range[3, 2000, 8], PrimeQ] (* _Jean-François Alcover_, Nov 17 2017 *)
%o A282721 (Python)
%o A282721 from sympy import isprime
%o A282721 def a(p):
%o A282721     r=(p - 1)//2
%o A282721     t=0
%o A282721     for j in range(1, r + 1):
%o A282721         q=(j**2)%p
%o A282721         if q<=r:t+=q
%o A282721     return t
%o A282721 print([a(p) for p in range(3, 2001, 8) if isprime(p)]) # _Indranil Ghosh_, Mar 27 2017, translated from Maple code
%Y A282721 Cf. A282035-A282043 and A282722-A282727.
%K A282721 nonn
%O A282721 1,2
%A A282721 _N. J. A. Sloane_, Feb 20 2017