A255644 -id:A255644 - cp's OEIS frontend

A228131 a(n) = Sum_{i=0..n-1} K(i,n)*i, where K(,) is Kronecker symbol.

0, 1, -1, 4, 0, 6, -7, 0, 27, 6, -11, -8, 0, 20, -30, 64, 0, -4, -19, 0, 0, 46, -69, -48, 250, 106, -9, 0, 0, 68, -93, 0, 0, 44, -70, 216, 0, 82, -156, 0, 0, 60, -43, -88, 0, 148, -235, -32, 1029, 94, -102, 0, 0, 6, -220, -224, 0, -82, -177, 0, 0, 168, -126, 1024, 0, 304, -67, 0, 0, 268, -497, 0, 0, 494, -50, -152, 0, 276, -395, 0, 2187, 4, -249, 0, 0, 310, -522, -176, 0, 388, -182, 0, 0, 424, -760, -192, 0, 202, 0, 2000

Offset: 1

Max Alekseyev, Aug 11 2013

sign

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).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Kronecker symbol

Cf. A255643, A255644

Table[Sum[KroneckerSymbol[k, n]*k, {k, 0, n - 1}], {n, 0, 50}] (* G. C. Greubel, Apr 23 2018 *)

PARI
```
a(n) = sum(i=1,n-1, kronecker(i,n)*i)
```

A373748 Triangle read by rows: T(n, k) is k if k is a quadratic residue modulo n, otherwise is -k and is a quadratic nonresidue modulo n. T(0, 0) = 0 by convention.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, -2, 3, 0, 1, -2, -3, 4, 0, 1, -2, -3, 4, 5, 0, 1, -2, 3, 4, -5, 6, 0, 1, 2, -3, 4, -5, -6, 7, 0, 1, -2, -3, 4, -5, -6, -7, 8, 0, 1, -2, -3, 4, -5, -6, 7, -8, 9, 0, 1, -2, -3, 4, 5, 6, -7, -8, 9, 10, 0, 1, -2, 3, 4, 5, -6, -7, -8, 9, -10, 11, 0, 1, -2, -3, 4, -5, -6, -7, -8, 9, -10, -11, 12

Offset: 0

Peter Luschny, Jun 27 2024

			Triangle starts:
  [0] [0]
  [1] [0,  1]
  [2] [0,  1,  2]
  [3] [0,  1, -2,  3]
  [4] [0,  1, -2, -3,  4]
  [5] [0,  1, -2, -3,  4,  5]
  [6] [0,  1, -2,  3,  4, -5,  6]
  [7] [0,  1,  2, -3,  4, -5, -6,  7]
  [8] [0,  1, -2, -3,  4, -5, -6, -7,  8]
  [9] [0,  1, -2, -3,  4, -5, -6,  7, -8,  9]
 [10] [0,  1, -2, -3,  4,  5,  6, -7, -8,  9,  10]

Carl Friedrich Gauss, Vierter Abschnitt. Von den Congruenzen zweiten Grades. Quadratische Reste und Nichtreste. Art. 97, in "Untersuchungen über die höhere Arithmetik", Hrsg. H. Maser, Verlag von Julius Springer, Berlin, 1889.
Peter Luschny, SageMath: is_quadratic_residue.

Signed version of A002262.
Cf. A000004 (column 0), A001477 (main diagonal), A255644(n) + n (row sums).
Cf. A096008, A096013, A373749.

QR := (a, n) -> ifelse(n = 0, 1, NumberTheory:-QuadraticResidue(a, n)):
for n from 0 to 10 do seq(a*QR(a, n), a = 0..n) od;

qr[n_] := qr[n] = Join[Table[PowerMod[k, 2, n], {k, 0, Floor[n/2]}], {n}];
T[0, 0] := 0; T[n_, k_] := If[MemberQ[qr[n], k], k, -k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

def Trow(n):
    q = set(mod(a * a, n) for a in range(n // 2  + 1)).union({n})
    return [k if k in q else -k for k in range(n + 1)]
for n in range(11): print(Trow(n))

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

Original entry on oeis.org

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

Max Alekseyev, Mar 01 2015

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

Cf. A228131, A255644.

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

{ 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 }

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

cp's OEIS Frontend

A228131 a(n) = Sum_{i=0..n-1} K(i,n)*i, where K(,) is Kronecker symbol.

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A373748 Triangle read by rows: T(n, k) is k if k is a quadratic residue modulo n, otherwise is -k and is a quadratic nonresidue modulo n. T(0, 0) = 0 by convention.

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

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Maple

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