A373749 -id:A373749 - cp's OEIS frontend

A373750 Middle terms of A373749.

0, 0, 1, 1, 0, 4, 3, 2, 0, 7, 5, 3, 0, 10, 7, 4, 0, 13, 9, 5, 0, 16, 11, 6, 0, 19, 13, 7, 0, 22, 15, 8, 0, 25, 17, 9, 0, 28, 19, 10, 0, 31, 21, 11, 0, 34, 23, 12, 0, 37, 25, 13, 0, 40, 27, 14, 0, 43, 29, 15, 0, 46, 31, 16, 0, 49, 33, 17, 0, 52, 35, 18, 0, 55

Offset: 0

Peter Luschny, Jun 23 2024

nonn

Cf. A373749.

REM := (n, k) -> ifelse(k = 0, n, irem(n, k)):
T := n -> local k; seq(REM(k^2, n), k = 0..n):
seq(T(n)[iquo(n, 2) + 1], n = 0..73);

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))

A378298 Number of solutions that satisfy the congruence: i^2 == j^2 (mod n) with 1 <= i < j <= n.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 3, 8, 6, 4, 5, 14, 6, 6, 15, 24, 8, 12, 9, 26, 22, 10, 11, 48, 20, 12, 27, 38, 14, 30, 15, 64, 36, 16, 41, 66, 18, 18, 43, 88, 20, 44, 21, 62, 72, 22, 23, 136, 42, 40, 57, 74, 26, 54, 67, 128, 64, 28, 29, 150, 30, 30, 105, 160, 80, 72, 33, 98

Offset: 1

Darío Clavijo, Nov 22 2024

nonn

a(n) >= A060594(n) for n >= 4.

			a(12) = 14 as the remainders 0 through 11 (mod 12) occur 2, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0 times respectively so a(12) = binomial(2, 2) + binomial(4, 2) + binomial(0, 2) + ... + binomial(0, 2) + binomial(0, 2) = 14. - _David A. Corneth_, Nov 25 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Darío Clavijo, Python program, Github.

Cf. A000079, A000217, A036289, A060594, A373749.

a:= n-> add(i*(i-1), i=coeffs(add(x^(j^2 mod n), j=1..n)))/2:
seq(a(n), n=1..68);  # Alois P. Heinz, Nov 25 2024

a[n_]:=Sum[Sum[Boole[PowerMod[i,2 , n ]== PowerMod[j,2 ,n]],{j,i+1,n}],{i,n}]; Array[a,68] (* Stefano Spezia, Nov 22 2024 *)

a(n) = sum(i=1, n, sum(j=i+1, n, Mod(i, n)^2 == Mod(j, n)^2)); \\ Michel Marcus, Nov 25 2024

a(n) = {
	my(v = vector(n), res = 0);
	for(i = 1, n,
		v[(i^2%n)+1]++;	
	);
	sum(i = 1, n, binomial(v[i], 2))
} \\ David A. Corneth, Nov 25 2024

from collections import defaultdict
def a(n: int) -> int:
    s = defaultdict(int)
    for i in range(1, n+1):
        s[pow(i,2,n)] += 1
    return sum(k*(k-1)>>1 for k in s.values())
print([a(n) for n in range(1, 69)])

from sympy import isprime
def A378298(n):
    if isprime(n): return n-1>>1
    c, d = [0]*n, 0
    for i in range(n):
        d += c[m:=i**2%n]
        c[m] += 1
    return d # Chai Wah Wu, Nov 28 2024

a(n) = Sum_{i=1..n} Sum_{j=i+1..n} [i^2 mod n == j^2 mod n], where [] denotes the Iverson bracket.
a(n) = Sum_{i=1..n} Sum_{j=i+1..n} [A373749(n,i) = A373749(n,j)] , where [] denotes the Iverson bracket.
a(2^k) = A036289(k-1).
If p is an odd prime, then a(p) = (p-1)/2. - Chai Wah Wu, Nov 27 2024

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A373750 Middle terms of A373749.

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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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A378298 Number of solutions that satisfy the congruence: i^2 == j^2 (mod n) with 1 <= i < j <= n.

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Maple

Mathematica

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