A262931 -id:A262931 - cp's OEIS frontend

A262932 Numbers k such that 7 is a square mod k.

1, 2, 3, 6, 7, 9, 14, 18, 19, 21, 27, 29, 31, 37, 38, 42, 47, 53, 54, 57, 58, 59, 62, 63, 74, 81, 83, 87, 93, 94, 103, 106, 109, 111, 113, 114, 118, 126, 131, 133, 137, 139, 141, 149, 159, 162, 166, 167, 171, 174, 177, 186, 189, 193, 197, 199, 203, 206, 217, 218, 222

Offset: 1

Erik Pelttari, Oct 04 2015

			7^2 == 7 (mod 14), so 14 is a term.
5^2 == 7 (mod 18) and 13^2 == 7 (mod 18), so 18 is a term.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A057125, A057126, A057762, A262931.

[n: n in [1..300] | exists(t){x : x in ResidueClassRing(n) | x^2 eq 7}]; // Vincenzo Librandi, Oct 05 2015

with(numtheory):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while mroot(7, 2, k)=FAIL do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 24 2017

Join[{1}, Table[If[Reduce[x^2 == 7, Modulus->n] === False, Null, n], {n, 2, 300}]//Union] (* Vincenzo Librandi, Oct 05 2015 *)

for(n=1, 300, if (issquare(Mod(7, n)), print1(n", "))); \\ Altug Alkan, Oct 04 2015

A381884 Triangle read by rows: T(n, k) = 0 if n = 0 or k is not a quadratic residue modulo n, otherwise T(n, k) = k.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Mar 17 2025

			Triangle T(n, k) starts:
  [0] 0;
  [1] 0, 1;
  [2] 0, 1, 2;
  [3] 0, 1, 0, 3;
  [4] 0, 1, 0, 0, 4;
  [5] 0, 1, 0, 0, 4, 5;
  [6] 0, 1, 0, 3, 4, 0, 6;
  [7] 0, 1, 2, 0, 4, 0, 0, 7;
  [8] 0, 1, 0, 0, 4, 0, 0, 0, 8;
  [9] 0, 1, 0, 0, 4, 0, 0, 7, 0, 9;
.
Array Arow(n) = [T(j, n), j = 0.. ] starts:
  [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  [1] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  [2] 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, ...
  [3] 0, 3, 3, 3, 0, 0, 3, 0, 0, 0, ...
  [4] 0, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
  [5] 0, 5, 5, 0, 5, 5, 0, 0, 0, 0, ...
  [6] 0, 6, 6, 6, 0, 6, 6, 0, 0, 0, ...
  [7] 0, 7, 7, 7, 0, 0, 7, 7, 0, 7, ...
  [8] 0, 8, 8, 0, 8, 0, 0, 8, 8, 0, ...
  [9] 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, ...
.
3 is not a quadratic residue modulo 7, therefore T(7, 3) = 0.
12 is a quadratic residue modulo 23, therefore T(23, 12) = 12.

Indices of the nonzero terms in row n of the array: A057126 (row 2), A057125 (row 3), A057762 (row 5), A262931 (row 6), A262932 (row 7).

QR := (k, n) -> ifelse(n = 0 or NumberTheory:-QuadraticResidue(k, n) < 0, 0, 1):
T := (n, k) -> k*QR(k, n): seq(seq(T(n, k), k = 0..n), n = 0..12);
Arow := (n, len) -> seq(T(j, n), j=0..len): seq(lprint([n], Arow(n, 9), n=0..9);

QR[n_, k_] := Module[{x, y}, If[Reduce[x^2 == n + k*y, {x, y}, Integers] =!= False, 1, -1]];
T[n_, k_] := If[n == 0 || QR[k, n] < 0, 0, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

from sympy.ntheory import is_quad_residue
def QR(n, k): return is_quad_residue(n, k)
def T(n, k): return 0 if n == 0 or not QR(k, n) else k
for n in range(13): print([T(n, k) for k in range(n + 1)])

cp's OEIS Frontend

A262932 Numbers k such that 7 is a square mod k.

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