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A255643 Difference between sums of quadratic residues and non-residues modulo n that are coprime to n.

%I A255643 #14 Feb 27 2025 10:49:16
%S A255643 0,1,-1,-2,0,-4,-7,-14,-3,0,-11,-22,0,0,-50,-44,0,-12,-19,-60,-84,-44,
%T A255643 -69,-94,0,0,-9,-98,0,-80,-93,-152,-176,0,-280,-138,0,-76,-312,-300,0,
%U A255643 -126,-43,-286,-330,0,-235,-332,-49,0,-476,-364,0,-36,-660,-602,-570,0,-177,-380,0,0,-630,-560,-780,-374,-67,-680,-782,-560,-497,-714,0,0,-850,-798,-1232,-468,-395,-1080,-27,0,-249,-882,-1360,-172,-1508,-1430,0,-600,-1820,-1058,-1674,0,-2090,-1240,0,0,-1518,-1100
%N A255643 Difference between sums of quadratic residues and non-residues modulo n that are coprime to n.
%H A255643 Robert Israel, <a href="/A255643/b255643.txt">Table of n, a(n) for n = 1..10000</a>
%F A255643 For prime n, a(n) = A228131(n) = A255644(n).
%F A255643 For prime n==1 (mod 4), a(n) = 0.
%F A255643 For prime n==3 (mod 4) and n > 3, i.e., n=A002145(m) for m > 1, a(n) = -n*A002143(m).
%F A255643 Is 2 the only n for which a(n) > 0? - _Robert Israel_, Feb 27 2025
%p A255643 f:= proc(n) local t;
%p A255643   add(`if`(igcd(t,n)=1, t*numtheory:-quadres(t,n),0), t=1..n-1)
%p A255643 end proc:
%p A255643 map(f, [$1..100]); # _Robert Israel_, Feb 27 2025
%o A255643 (PARI) { A255643(n) = my(r); r=0; for(i=0,n-1, if(gcd(i,n)>1,next); if(issquare(Mod(i,n)), r+=i, r-=i) ); r }
%Y A255643 Cf. A228131, A255644.
%K A255643 sign,look
%O A255643 1,4
%A A255643 _Max Alekseyev_, Mar 01 2015