A057125 -id:A057125 - cp's OEIS frontend

A057759 Least nonnegative square root of 3 mod n for n in A057125.

0, 1, 0, 3, 5, 4, 5, 7, 9, 6, 15, 9, 7, 12, 11, 8, 27, 30, 28, 21, 15, 9, 13, 35, 10, 18, 49, 15, 11, 27, 53, 38, 39, 12, 43, 17, 21, 72, 13, 62, 61, 48, 19, 33, 69, 24, 14, 87, 99, 89, 49, 21, 15, 50, 71, 106, 56, 27, 96, 76, 16, 93, 23, 130

Offset: 1

N. J. A. Sloane, Nov 01 2000

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A057125.

A057125 = Prepend[ Select[ Range[300], Reduce[ Mod[3 - k^2, #] == 0, k, Integers] =!= False &], 1]; a[n_] := Min[ r /. {ToRules[ Reduce[ Mod[r^2 - 3, A057125[[n]]] == 0, r, Integers] /. C[1] -> 0]}]; a[1] = 0; Table[a[n], {n, 1, 64}] (* Jean-François Alcover, Sep 20 2012 *)

Corrected by T. D. Noe, Apr 19 2007 [The errors were caused by the faulty Maple command "mroot"]

A057758 Duplicate of A057125.

Original entry on oeis.org

1, 2, 3, 6, 11, 13, 22, 23, 26, 33, 37, 39, 46, 47, 59, 61, 66, 69, 71, 73, 74, 78, 83

Offset: 1

dead

A057126 Numbers k such that 2 is a square mod k.

Original entry on oeis.org

1, 2, 7, 14, 17, 23, 31, 34, 41, 46, 47, 49, 62, 71, 73, 79, 82, 89, 94, 97, 98, 103, 113, 119, 127, 137, 142, 146, 151, 158, 161, 167, 178, 191, 193, 194, 199, 206, 217, 223, 226, 233, 238, 239, 241, 254, 257, 263, 271, 274, 281, 287, 289, 302, 311, 313, 322

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

Numbers that are not multiples of 4 and for which all odd prime factors are congruent to +/- 1 mod 8. - Eric M. Schmidt, Apr 20 2013
Apparently the same as the list of numbers primitively represented by the indefinite quadratic form x^2 - 2y^2 (cf. A035251). - N. J. A. Sloane, Jun 11 2014
From Wolfdieter Lang, Jul 11 2025: (Start)
Also the negative sequence lists the numbers properly represented by the indefinite quadratic form x^2 - 2*y^2  of discriminant 4*2 = 8. For the proof see the W. Lang paper linked in A385449, Lemma 18, pp. 22-23.
The connection between the proper positive fundamental solutions (X, Y) of X^2 - 2*Y^2 = -a(n), given in A385449, and the solutions (x, y) of x^2 - 2*y^2 = a(n) is (x, y) = (2*Y - X, X - Y). If y becomes nonpositive a transformation with the matrix Mat([3,4], [2,3]) will give the positive proper fundamental solution. See the example section of A385449. See also the Nov 09 2009 comment in A035251 by Franklin T. Adams-Watters for this connection, and for the matrix eq. (38) p. 14 of the mentioned linked paper.
Therefore the previous statement on the representation of a(n) is true.(End)

T. D. Noe, 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)

Includes the primes in A038873 and these (primes congruent to {1, 2, 7} mod 8) are the prime factors of the terms in this sequence.
Cf. A008784, A057125, A057127, A057128, A057129, A057757, A035251, A038873.
Cf. A087780 (number of solutions mod n).

with(numtheory); [seq(mroot(2,2,p),p=1..300)];

ok[n_] := Reduce[ Mod[2 - k^2, n] == 0, k, Integers] =!= False; Prepend[ Select[ Range[400], ok], 1] (* Jean-François Alcover, Sep 20 2012 *)

isok(n) = issquare(Mod(2,n)); \\ Michel Marcus, Feb 19 2016

Checked by T. D. Noe, Apr 19 2007

A057127 -2 is a square mod n.

Original entry on oeis.org

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

Henry Bottomley, Aug 10 2000

nonn

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

			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

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)

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

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

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

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

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

A038874 Primes p such that 3 is a square mod p.

Original entry on oeis.org

2, 3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601

Offset: 1

N. J. A. Sloane

Also primes congruent to {1, 2, 3, 11} mod 12.
The subsequence p = 1 (mod 4) corresponds to A068228 and only these entries of a(n) are squares mod 3 (from the quadratic reciprocity law). - Lekraj Beedassy, Jul 21 2004
Largest prime factors of n^2 - 3. - Vladimir Joseph Stephan Orlovsky, Aug 12 2009
Aside from 2 and 3, primes p such that Legendre(3, p) = 1. Bolker asserts there are infinitely many of these primes. - Alonso del Arte, Nov 25 2015
The associated bases of the squares are 1, 0, 5, 4, 7, 15, 12, 11, 8, 28, 21, 13...: 1^2 = 3 -1*2, 0^2 = 3-1*3, 5^2 = 3+ 2*11, 4^2 = 3+1*13, 7^2 = 3+2*23, 15^2 = 3+6*37, 12^2 = 3+3*47,... - R. J. Mathar, Feb 23 2017

			11 is in the sequence since the equation x^2 - 11y = 3 has solutions, such as x = 5, y = 2.
13 is in the sequence since the equation x^2 - 13y = 3 has solutions, such as x = 4, y = 1.
17 is not in the sequence because x^2 - 17y = 3 has no solutions in integers; Legendre(3, 17) = -1.

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 74, Theorem 25.3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.

Cf. A002313, A033203, A038873, A045331, A057125.

If the first two terms are omitted we get A097933. A040101 is another sequence.

[p: p in PrimesUpTo(1200) | p mod 12 in [1, 2, 3, 11]]; // Vincenzo Librandi, Aug 08 2012

select(isprime, [2,3, seq(seq(6+s+12*i, s=[-5,5]),i=0..1000)]); # Robert Israel, Dec 23 2015

Select[Prime[Range[250]], MemberQ[{1, 2, 3, 11}, Mod[#, 12]] &] (* Vincenzo Librandi, Aug 08 2012 *)
Select[Flatten[Join[{2, 3}, Table[{12n - 1, 12n + 1}, {n, 50}]]], PrimeQ] (* Alonso del Arte, Nov 25 2015 *)

forprime(p=2, 1e3, if(issquare(Mod(3, p)), print1(p , ", "))) \\ Altug Alkan, Dec 04 2015

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

More terms from Henry Bottomley, Aug 10 2000

A057128 Numbers n such that -3 is a square mod n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 12, 13, 14, 19, 21, 26, 28, 31, 37, 38, 39, 42, 43, 49, 52, 57, 61, 62, 67, 73, 74, 76, 78, 79, 84, 86, 91, 93, 97, 98, 103, 109, 111, 114, 122, 124, 127, 129, 133, 134, 139, 146, 147, 148, 151, 156, 157, 158, 163, 169, 172, 181, 182, 183, 186, 193

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

The fact that there are no numbers in this sequence of the form 6k+5 leads to the result that all prime factors of central polygonal numbers (A002061 of the form n^2-n+1) are either 3 or of the form 6k+1. This in turn leads to there being an infinite number of primes of the form 6k+1, since if P=product[all known primes of form 6k+1] then all the prime factors of 9P^2-3P+1 must be unknown primes of form 6k+1.
Numbers that are not multiples of 8 or 9 and for which all prime factors greater than 3 are congruent to 1 mod 6. - Eric M. Schmidt, Apr 21 2013
Numbers that divide at least some member of A117950. - Robert Israel, Feb 19 2016

			a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Includes the primes in A045331 and these (primes congruent to {1, 2, 3} mod 6) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057129.

Cf. A117950.

select(t -> numtheory:-quadres(-3,t) = 1, {$1..1000}); # Robert Israel, Feb 19 2016

Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 05 2019 *)

isok(n) = issquare(Mod(-3,n)); \\ Michel Marcus, Feb 19 2016

def A057128(n) :
    if n%8==0 or n%9==0: return False
    for (p, m) in factor(n) :
        if p % 6 not in [1, 2, 3] : return False
        return True
# Eric M. Schmidt, Apr 21 2013

A057129 -4 is a square mod n.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, 74, 82, 85, 89, 97, 100, 101, 104, 106, 109, 113, 116, 122, 125, 130, 136, 137, 145, 146, 148, 149, 157, 164, 169, 170, 173, 178, 181, 185, 193, 194, 197, 200, 202, 205, 212

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

Numbers that are not multiples of 16 and for which all odd prime factors are congruent to 1 mod 4. - Eric M. Schmidt, Apr 21 2013

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Includes the primes in A002313 and these (primes congruent to {1, 2} mod 4) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057128.

Select[Range[100], IntegerQ[PowerMod[-4, 1/2, #]] &] // Quiet (* After Jean-François Alcover *) (* Robert Price, Apr 19 2025 *)

def A057129(n) :
    if n%16==0: return False
    for (p, m) in factor(n) :
        if p % 4 not in [1, 2] : return False
    return True
# Eric M. Schmidt, Apr 21 2013

A057762 Numbers k such that 5 is a square mod k.

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 19, 20, 22, 29, 31, 38, 41, 44, 55, 58, 59, 61, 62, 71, 76, 79, 82, 89, 95, 101, 109, 110, 116, 118, 121, 122, 124, 131, 139, 142, 145, 149, 151, 155, 158, 164, 178, 179, 181, 190, 191, 199, 202, 205, 209, 211, 218, 220, 229

Offset: 1

N. J. A. Sloane, Nov 01 2000

nonn

Numbers not divisible by 3, 8, or 25 and whose prime factors > 5 are congruent to +/- 1 mod 5. - Eric M. Schmidt, Jan 24 2014

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A057126, A057125, A057763.

with(numtheory); [seq(mroot(5,2,p),p=1..400)];

Prepend[ Select[ Range[300], Reduce[ Mod[5 - k^2, #] == 0, k, Integers] =!= False &], 1] (* Jean-François Alcover, Sep 20 2012 *)
Join[{1, 2, 4, 5}, Select[Range[6, 300], MemberQ[Mod[Range[#]^2, #], 5] &]] (* T. D. Noe, Sep 20 2012 *)

A262931 Numbers k such that 6 is a square mod k.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 15, 19, 23, 25, 29, 30, 38, 43, 46, 47, 50, 53, 57, 58, 67, 69, 71, 73, 75, 86, 87, 94, 95, 97, 101, 106, 114, 115, 125, 129, 134, 138, 139, 141, 142, 145, 146, 149, 150, 159, 163, 167, 173, 174, 190, 191, 193, 194, 197, 201, 202, 211, 213

Offset: 1

Erik Pelttari, Oct 04 2015

			6^2 == 6 (mod 15), so 15 is a term.

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

Cf. A057125, A057126, A057762, A262932.

[n: n in [1..300] | exists(t){x : x in ResidueClassRing(n) | x^2 eq 6}]; // 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(6, 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 == 6, Modulus->n] === False, Null, n], {n, 2, 300}]//Union] (* Vincenzo Librandi, Oct 05 2015 *)

for(n=1, 300, if (issquare(Mod(6, n)), print1(n", "))); \\ Altug Alkan, Oct 04 2015

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

Original entry on oeis.org

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

cp's OEIS Frontend

A057759 Least nonnegative square root of 3 mod n for n in A057125.

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A057758 Duplicate of A057125.

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A057126 Numbers k such that 2 is a square mod k.

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A057127 -2 is a square mod n.

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A038874 Primes p such that 3 is a square mod p.

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A057128 Numbers n such that -3 is a square mod n.

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A057129 -4 is a square mod n.

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A057762 Numbers k such that 5 is a square mod k.

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