cp's OEIS Frontend

A282040 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.

%I A282040 #17 Aug 22 2025 15:06:14
%S A282040 4,59,126,285,679,953,1706,2675,3709,4269,5551,6480,8488,8858,11194,
%T A282040 12212,15103,20665,23511,24153,30197,32733,38458,36913,42643,42032,
%U A282040 59638,64987,70396,70887,85606,94192,95522,99930,123090,117932,130367,134436,141262,149395,169769,167663,175469
%N A282040 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p that are > p/2.
%H A282040 Robert Israel, <a href="/A282040/b282040.txt">Table of n, a(n) for n = 1..10000</a>
%H A282040 Christian Aebi and Grant Cairns. <a href="http://arxiv.org/abs/1512.00896">Sums of Quadratic residues and nonresidues</a>, arXiv preprint arXiv:1512.00896 [math.NT],  2015.
%p A282040 with(numtheory):
%p A282040 Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
%p A282040 for i1 from 1 to 300 do
%p A282040 p:=ithprime(i1);
%p A282040 if (p mod 8) = 7 then
%p A282040 ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
%p A282040 for j from 1 to p-1 do
%p A282040 if legendre(j,p)=1 then
%p A282040 q:=q+j;
%p A282040 if j<p/2 then ql:=ql+j; else qu:=qu+j; fi;
%p A282040 else
%p A282040 n:=n+j;
%p A282040 if j<p/2 then nl:=nl+j; else nu:=nu+j; fi;
%p A282040 fi;
%p A282040 od;
%p A282040 Ql:=[op(Ql),ql];
%p A282040 Qu:=[op(Qu),qu];
%p A282040 Q:=[op(Q),q];
%p A282040 Nl:=[op(Nl),nl];
%p A282040 Nu:=[op(Nu),nu];
%p A282040 N:=[op(N),n];
%p A282040 fi;
%p A282040 od:
%p A282040 Ql; Qu; Q; Nl; Nu; N; # A282039, A282040, A282041, A282039 again, A282042, A282043
%p A282040 # alternative:
%p A282040 g:= proc(t, p) if t > p/2 then t else 0 fi end proc;
%p A282040 f:= proc(n) local k;
%p A282040      add(g(k^2 mod n, n), k=1..n/2)
%p A282040 end proc:
%p A282040 P:= select(isprime, [seq(i, i=7..3000, 8)]):
%p A282040 map(f, P); # _Robert Israel_, Aug 22 2025
%t A282040 sum[p_]:= Total[If[#>p/2 && JacobiSymbol[#, p] == 1, #, 0]& /@ Range[p-1]];
%t A282040 sum /@ Select[Range[7, 1000, 8], PrimeQ] (* _Jean-François Alcover_, Aug 31 2018 *)
%Y A282040 Cf. A282035-A282043 and A282721-A282727.
%K A282040 nonn
%O A282040 1,1
%A A282040 _N. J. A. Sloane_, Feb 20 2017