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

0, 1, -1, -2, 0, -4, -7, -14, -3, 0, -11, -22, 0, 0, -50, -44, 0, -12, -19, -60, -84, -44, -69, -94, 0, 0, -9, -98, 0, -80, -93, -152, -176, 0, -280, -138, 0, -76, -312, -300, 0, -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

Offset: 1

Author

Max Alekseyev, Mar 01 2015

Keywords

• sign
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A228131, A255644.

Programs

Maple

f:= proc(n) local t;
  add(`if`(igcd(t,n)=1, t*numtheory:-quadres(t,n),0), t=1..n-1)
end proc:
map(f, [$1..100]); # Robert Israel, Feb 27 2025

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 }

Formula

For prime n, a(n) = A228131(n) = A255644(n).

For prime n==1 (mod 4), a(n) = 0.

For prime n==3 (mod 4) and n > 3, i.e., n=A002145(m) for m > 1, a(n) = -n*A002143(m).

Is 2 the only n for which a(n) > 0? - Robert Israel, Feb 27 2025