A282722 - OEIS

%I #16 Nov 27 2017 10:41:39

%S 0,9,44,293,461,758,1022,1799,2530,3171,4778,5068,7662,8470,9993,

%T 14097,16674,19467,20755,25701,29042,34471,37506,40661,45066,48541,

%U 54324,54224,58135,60351,68593,75432,74014,83881,85900,98518,112000,117443,122241,132125,143322,151299,163180,161975,181191

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

%H Jean-François Alcover, <a href="/A282722/b282722.txt">Table of n, a(n) for n = 1..1000</a>

%H Christian Aebi, Grant Cairns, <a href="http://arxiv.org/abs/1512.00896">Sums of Quadratic residues and nonresidues</a>, arXiv:1512.00896 [math.NT], 2015.

%p with(numtheory):

%p Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];

%p for i1 from 1 to 300 do

%p p:=ithprime(i1);

%p if (p mod 8) = 3 then

%p ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;

%p for j from 1 to p-1 do

%p if legendre(j,p)=1 then

%p q:=q+j;

%p if j<p/2 then ql:=ql+j; else qu:=qu+j; fi;

%p else

%p n:=n+j;

%p if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;

%p fi;

%p od;

%p Ql:=[op(Ql),ql];

%p Qu:=[op(Qu),qu];

%p Q:=[op(Q),q];

%p Nl:=[op(Nl),nl];

%p Nu:=[op(Nu),nu];

%p N:=[op(N),n];

%p Th:=[op(Th),q+ql];

%p fi;

%p od:

%p Ql; Qu; Q; Nl; Nu; N; Th; # A282721 - A282727

%p # 2nd program

%p A282722 := proc(n)

%p     local p,ar;

%p     p := A007520(n) ;

%p     a := 0 ;

%p     for r from (p+1)/2 to p do

%p         if numtheory[legendre](r,p) = 1 then

%p             a := a+r ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A282722(n),n=1..10) ; # _R. J. Mathar_, Apr 07 2017

%t b[1] = 3; b[n_] := b[n] = Module[{p}, p = NextPrime[b[n - 1]]; While[Mod[p, 8] != 3, p = NextPrime[p]]; p];

%t a[n_] := Module[{p, q, r}, p = b[n]; q = 0; For[r = (p + 1)/2, r <= p, r++, If[KroneckerSymbol[r, p] == 1, q = q + r]]; q];

%t Array[a, 45] (* _Jean-François Alcover_, Nov 27 2017, after _R. J. Mathar_ *)

%Y Cf. A282035-A282043 and A282721-A282727.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Feb 20 2017