A163183 -id:A163183 - cp's OEIS frontend

A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

2, 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 547, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 659, 673, 683

Offset: 1

N. J. A. Sloane

Sequence naturally partitions into two sequences: all primes p with ord_p(-2) odd (A163183, the primes dividing 2^j +1 for some odd j) and certain primes p with ord_p(-2) even (A163185). - Christopher J. Smyth, Jul 23 2009

Terms m in A047476 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002332, A002333.
Cf. A039706, A003628 (complement with respect to A000040).
Primes in A002479.
Cf. A051100 (see Mathar's comment).
Apart from leading term the same as A033200.

a033203 n = a033203_list !! (n-1)
a033203_list = filter ((== 1) . a010051) a047476_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1..3]]; // Vincenzo Librandi, Aug 11 2012

[p: p in PrimesUpTo(800) | NormEquation(2,p) eq true]; // Bruno Berselli, Jul 03 2016

QuadPrimes2[1, 0, 2, 10000] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,8]]&] (* Harvey P. Dale, Mar 16 2013 *)

is(n)=isprime(n) && issquare(Mod(-2,n)) \\ Charles R Greathouse IV, Nov 29 2016

a(n) = A002332(n) + 2*A002333(n)^2. - Zak Seidov, May 29 2014

A014663 Primes p such that multiplicative order of 2 modulo p is odd.

Original entry on oeis.org

7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 263, 271, 311, 337, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 601, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919, 937, 967, 983, 991, 1031, 1039, 1063

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Or, primes p which do not divide 2^n+1 for any n.
The possibility n=0 in the above rules out A072936(1)=2; apart from this, a(n)=A072936(n+1). - M. F. Hasler, Dec 08 2007
The order of 2 mod p is odd iff 2^k=1 mod p, where p-1=2^s*k, k odd. - M. F. Hasler, Dec 08 2007
Has density 7/24 (Hasse).
From Jianing Song, Jun 27 2025: (Start)
The multiplicative order of 2 modulo a(n) is A139686(n).
Contained in primes congruent to 1 or 7 modulo 8 (primes p such that 2 is a quadratic residue modulo p, A001132), and contains primes congruent to 7 modulo 8 (A007522). (End)

Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.
P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
H. H. Hasse, Über die Dichte der Primzahlen p, ... , Math. Ann., 168 (1966), 19-23.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
Chunlei Li, Nian Li, and Matthew G. Parker, Complementary Sequence Pairs of Types II and III. [From _N. J. A. Sloane_, Jun 16 2012]

Cf. Complement in primes of A091317.
Cf. A001132, A007522, A040098, A045315, A049564, A139686 (the actual multiplicative orders).
Cf. Essentially the same as A072936 (except for missing leading term 2).
Cf. other bases: this sequence (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

okQ[p_] := OddQ[MultiplicativeOrder[2, p]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)

isA014663(p)=1==Mod(1,p)<<((p-1)>>factor(p-1,2)[1,2])
listA014663(N=1000)=forprime(p=3,N,isA014663(p)&print1(p", ")) \\ M. F. Hasler, Dec 08 2007

lista(nn) = {forprime(p=3, nn, if (znorder(Mod(2, p)) % 2, print1(p, ", ")););} \\ Michel Marcus, Feb 06 2015

Edited by M. F. Hasler, Dec 08 2007

More terms from Max Alekseyev, Feb 06 2010

A385192 Primes p such that multiplicative order of 5 modulo p is odd.

Original entry on oeis.org

2, 11, 19, 31, 59, 71, 79, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 239, 251, 269, 271, 311, 331, 359, 379, 389, 401, 409, 419, 431, 439, 461, 479, 491, 499, 541, 569, 571, 599, 619, 631, 659, 691, 719, 739, 751, 811, 829, 839, 859, 911, 919, 941, 971, 991

Offset: 1

Jianing Song, Jun 20 2025

The multiplicative order of 5 modulo a(n) is A385193(n).
Contained in primes congruent to 1 or 4 modulo 5 (primes p such that 5 is a quadratic residue modulo p, A045468), and contains primes congruent to 11 or 19 modulo 20 (A122869).
Conjecture: this sequence has density 1/3 among the primes.

			101 is a term since 5^25 == 1 (mod 101).

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A040105, which (without the terms 2 and 5) is itself a subsequence of A045468.
Contains A122869 as a proper subsequence.
Cf. A385193 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), this sequence (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[5, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385192(p) = isprime(p) && (p!=5) && znorder(Mod(5,p))%2

A385220 Primes p such that multiplicative order of 3 modulo p is odd.

Original entry on oeis.org

2, 11, 13, 23, 47, 59, 71, 83, 107, 109, 131, 167, 179, 181, 191, 227, 229, 239, 251, 263, 277, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 587, 599, 601, 647, 659, 683, 709, 719, 733, 743, 757, 827, 829, 839, 863, 887, 911, 947, 971, 983

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of 3 modulo a(n) is A385226(n).
Without 2, contained in primes congruent to 1 or 11 modulo 12 (primes p such that 3 is a quadratic residue modulo p; A097933), and contains primes congruent to 11 modulo 12 (A068231).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

A068231 < this sequence < A045317 < A040101 < A097933 (ignoring terms 2, 3), where Ax < Ay means that Ax is a subsequence of Ay.
Complement of A301916 in {primes} \ {3}.
Cf. A385226 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), this sequence (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[3, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385220(p) = isprime(p) && (p!=3) && znorder(Mod(3,p))%2

A385224 Primes p such that multiplicative order of -4 modulo p is odd.

Original entry on oeis.org

5, 13, 29, 37, 41, 53, 61, 101, 109, 113, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 313, 317, 349, 373, 389, 397, 409, 421, 457, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 761, 773, 797, 809, 821, 829, 853, 857, 877, 941, 953, 997

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of -4 modulo a(n) is A385230(n).
Different from A133204: 593 is here but not in A133204, and 1601 is in A133204 but not here.
The sequence contains no primes congruent to 3 modulo 4 and all primes congruent to 5 modulo 8:
  - If p is a term of this sequence, then -4 is a quadratic residue modulo p, so p == 1 (mod 4);
  - For p == 1 (mod 4), we have (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i is a solution to i^2 == -1 (mod p).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A002144 (primes congruent to 1 modulo 4).
Contains A007521 (primes congruent to 5 or modulo 8) as a proper subsequence.
Cf. A385230 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), this sequence (base -4), A385225 (base -5).
Cf. A133204.

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-4, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385224(p) = isprime(p) && (p!=2) && znorder(Mod(-4,p))%2

A385221 Primes p such that multiplicative order of 4 modulo p is odd.

Original entry on oeis.org

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 73, 79, 83, 89, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 233, 239, 251, 263, 271, 281, 283, 307, 311, 331, 337, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of 4 modulo a(n) is A385227(n).
Primes p such that neither ord(2,p) nor ord(-2,p) is divisible by 4, where ord(a,m) is the multiplicative order of a modulo m. (Note that we have either (a) ord(2,p) = ord(-2,p) and both are even; (b) ord(-2,p) = 2*ord(2,p), ord(2,p) is odd, ord(-2,p) == 2 (mod 4); or (c) ord(2,p) = 2*ord(-2,p), ord(-2,p) is odd, ord(2,p) == 2 (mod 4)).
Contains all primes congruent to 3 modulo 4 (A002145).
Conjecture: this sequence has density 7/12 among the primes (see A014663).

Jianing Song, Table of n, a(n) for n = 1..10000

Contains A002145, A014663, and A163183.
Cf. A385227 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), this sequence (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[4, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385221(p) = isprime(p) && (p!=2) && znorder(Mod(4,p))%2

A385223 Primes p such that multiplicative order of -3 modulo p is odd.

Original entry on oeis.org

2, 7, 19, 31, 37, 43, 61, 67, 79, 103, 127, 139, 151, 157, 163, 199, 211, 223, 271, 283, 307, 331, 349, 367, 373, 379, 397, 439, 463, 487, 499, 523, 547, 571, 607, 613, 619, 631, 643, 661, 691, 727, 739, 751, 787, 811, 823, 853, 859, 877, 883, 907, 919, 937, 967, 991, 997

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of -3 modulo a(n) is A385229(n).
Without 2, contained in primes congruent to 1 modulo 3 (primes p such that -3 is a quadratic residue modulo p, A002476), and contains primes congruent to 7 modulo 12 (A068229).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A002476. Contains A068229 as a subsequence.
Cf. A385229 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), this sequence (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-3, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385223(p) = isprime(p) && (p!=3) && znorder(Mod(-3,p))%2

A385225 Primes p such that multiplicative order of -5 modulo p is odd.

Original entry on oeis.org

2, 3, 7, 23, 29, 43, 47, 61, 67, 83, 103, 107, 127, 163, 167, 223, 227, 229, 263, 283, 307, 347, 349, 367, 383, 421, 443, 449, 463, 467, 487, 503, 509, 521, 523, 547, 563, 587, 607, 643, 647, 661, 683, 701, 709, 727, 743, 761, 787, 821, 823, 827, 863, 883, 887, 907, 947, 967, 983

Offset: 1

Jianing Song, Jun 22 2025

The multiplicative order of -5 modulo a(n) is A385231(n).
Contained in primes congruent to 1, 3, 7, 9 modulo 20 (primes p such that -5 is a quadratic residue modulo p, A139513), and contains primes congruent to 3, 7 modulo 20 (A122870).
Conjecture: this sequence has density 1/3 among the primes.

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A139513. Contains A122870 as a subsequence.
Cf. A385231 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), this sequence (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-5, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385225(p) = isprime(p) && (p!=5) && znorder(Mod(-5,p))%2

A385228 Odd multiplicative orders of -2 modulo primes.

Original entry on oeis.org

1, 5, 9, 7, 29, 33, 41, 53, 65, 69, 81, 89, 105, 113, 25, 35, 47, 51, 15, 173, 189, 209, 221, 233, 245, 83, 261, 273, 281, 57, 293, 77, 309, 107, 329, 11, 115, 123, 393, 135, 413, 429, 441, 453, 473, 97, 509, 129, 131, 175, 545, 137, 561, 83, 585, 593, 149, 629, 641, 645, 653, 713, 725

Offset: 1

Jianing Song, Jun 22 2025

a(n) is the multiplicative order of -2 modulo A163183(n).

Odd elements in A337878 (with first term changed to 1).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A337878, A163183 (corresponding primes).

Cf. other bases: A139686 (base 2), A385226 (base 3), A385227 (base 4), A385193 (base 5), this sequence (base -2), A385229 (base -3), A385230 (base -4), A385231 (base -5).

Select[Map[MultiplicativeOrder[-2, #] &, Prime[Range[250]]], OddQ] (* Paolo Xausa, Jun 28 2025 *)

forprime(p=3, 1e4, z=znorder(Mod(-2, p)); if(z%2, print1(z, ", ")))

A051085 Primes p such that x^32 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 281, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 617, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1033, 1049, 1051, 1091, 1097, 1123, 1163, 1171

Offset: 1

N. J. A. Sloane

Complement of A216747 relative to A000040. - Vincenzo Librandi, Sep 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A163183.

[p: p in PrimesUpTo(1200) | exists(t){x : x in ResidueClassRing(p) | x^32 eq - 2}]; // Vincenzo Librandi, Sep 15 2012

ok[p_]:= Reduce[Mod[x^32 + 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[400]], ok] (* Vincenzo Librandi, Sep 15 2012 *)

forprime(p=2, 2000, if([]~!=polrootsmod(x^32+2, p), print1(p, ", "))); print();
/* Joerg Arndt, Jun 24 2012 */

More terms from Joerg Arndt, Jul 27 2011

cp's OEIS Frontend

A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

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A014663 Primes p such that multiplicative order of 2 modulo p is odd.

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A385192 Primes p such that multiplicative order of 5 modulo p is odd.

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A385220 Primes p such that multiplicative order of 3 modulo p is odd.

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A385224 Primes p such that multiplicative order of -4 modulo p is odd.

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A385221 Primes p such that multiplicative order of 4 modulo p is odd.

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A385223 Primes p such that multiplicative order of -3 modulo p is odd.

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