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

%I #13 Nov 27 2017 11:35:24

%S 0,2,13,94,129,247,306,555,745,999,1579,1555,2466,2653,3059,4581,5430,

%T 6351,6658,8409,9087,11158,11996,12858,14814,15788,17880,17277,18950,

%U 19481,22400,24876,23518,27448,28115,32285,36743,38269,39851,43111,47406,50055,53683,51645,58274,66410,65119,76013,80465

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

%H Jean-François Alcover, <a href="/A282724/b282724.txt">Table of n, a(n) for n = 1..1000</a>

%H Aebi, Christian, and Grant Cairns. <a href="http://arxiv.org/abs/1512.00896">Sums of Quadratic residues and nonresidues</a>, arXiv preprint arXiv:1512.00896 (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 A282724 := proc(n)

%p local p,a,r;

%p p := A007520(n) ;

%p a := 0 ;

%p for r from 1 to (p-1)/2 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(A282724(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 = 1, r <= (p - 1)/2, r++, If[KroneckerSymbol[r, p] != 1, q = q + r]]; q];

%t Array[a, 50] (* _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