A057127 -2 is a square mod n.

1, 2, 3, 6, 9, 11, 17, 18, 19, 22, 27, 33, 34, 38, 41, 43, 51, 54, 57, 59, 66, 67, 73, 81, 82, 83, 86, 89, 97, 99, 102, 107, 113, 114, 118, 121, 123, 129, 131, 134, 137, 139, 146, 153, 162, 163, 166, 171, 177, 178, 179, 187, 193, 194, 198, 201, 209, 211, 214, 219

Offset: 1

Author

Henry Bottomley, Aug 10 2000

Keywords

• nonn

Comments

Includes the primes in A033203 and these (primes congruent to {1, 2, 3} mod 8) are the prime factors of the terms in this sequence.

Numbers that are not multiples of 4 and for which all odd prime factors are congruent to {1, 3} mod 8. - Eric M. Schmidt, Apr 21 2013

Positive integers primitively represented by x^2 + 2y^2. - Ray Chandler, Jul 22 2014

The set of the divisors of numbers of the form k^2+2. - Michel Lagneau, Jun 28 2015

The number of proper solutions (x, y) with nonnegative x of the positive definite primitive quadratic form x^2 + 2*y*2 (discriminant -8) representing a(n) is 1 for n = 1 and for n >= 2 it is 2^(P_1 + P_3), where P_1 and P_3 are the number of distinct prime divisors of a(n) congruent to 1 and 3 modulo 8, respectively. See the above comments on A033203 and this binary form. - Wolfdieter Lang, Feb 25 2021

Examples

			Binary quadratic form x^2 + 2*y^2 representing a(n), with x >= 0: a(1) = 1: one solution (x, y) = (1,0); a(2) = 2: one solution (0,1); a(3) = 3: two solutions (1, pm 1), with pm = +1 or -1; a(5) = 9 = 3^2: two solutions (1, pm 2); a(12) = 33 = 3*11: 4 solutions (1, pm 4) and (5, pm 2); a(137) = 3*11*17 = 561: eight solutions (7, pm 16), (13, pm 14), (19, pm 10) and (23, pm 4). - _Wolfdieter Lang_, Feb 25 2021
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A008784, A033203, A057125, A057126, A057128, A057129.

Programs

Maple

select(n -> numtheory:-msqrt(-2,n) <> FAIL, [$1..1000]); # Robert Israel, Jun 29 2015

Mathematica

Select[Range[300], IntegerQ[PowerMod[-2, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 04 2019 *)

PARI

isok(n) = issquare(Mod(-2, n)); \\ Michel Marcus, Jun 28 2015

Sage

def isA057127(n):
    if n % 4 == 0: return False
    return all(p % 8 in [1, 2, 3] for p, _ in factor(n))
[n for n in range(1, 300) if isA057127(n)]
# Eric M. Schmidt, Apr 21 2013