A385228 -id:A385228 - cp's OEIS frontend

A163183 Primes dividing 2^j + 1 for some odd j.

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

Offset: 1

Christopher J. Smyth, Jul 22 2009

Also the primes p for which ord_p(-2) is odd, as (-2)^j == 1 (mod p).
All such p are = 1 or 3 mod 8, so sequence is subsequence of A033200, as (-2)^{j+1} == -2 (mod p) implies that (-2/p) = 1, p == 1 or 3 (mod 8).
Claim: Sequence contains all primes = 3 mod 8, so contains A007520 as a subsequence.
Proof: If p = 8r + 3 then 2^{4r+1} == 1 or -1 (mod p). If former, then (2^{2r+1})^2 == 2 (mod p), (2/p) = 1, only true for p == 1 or 7 (mod 8). So p | 2^{4r+1} + 1.
Also contains some primes == 1 (mod 8), given in A163184. So sequence is a union of A007520 and A163184.
Claim: For every p in sequence and every 2^k, the equation x^{2^k} == -2 (mod p) is soluble. Hence sequence is a subsequence of A033203 (k=1), A051071 (k=2), A051073 (k=3), A051077 (k=4), A051085 (k=5), A051101 (k=6), ....
Proof: Put x == (-2)^u (mod p). Then using (-2)^j == 1 (mod p), we can solve x^{2^k} == -2 (mod p) if can find u and v such that u*2^k + v*j = 1, possible as gcd(2^k, j) = 1.
From Jianing Song, Jun 22 2025: (Start)
The multiplicative order of -2 modulo a(n) is A385228(n).
Contained in primes congruent to 1 or 3 modulo 8 (primes p such that -2 is a quadratic residue modulo p, A033200), and contains primes congruent to 3 modulo 8 (A007520).
Conjecture: this sequence has density 7/24 among the primes (see A014663). (End)

			11 is in sequence as 11 | 2^5 + 1; 281 (smallest element of the sequence == 1 (mod 8)) is in the sequence as 281 | 2^35 + 1.

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

Sequence is a union of A007520 and A163184.
Subsequence of A033200. Contains A007520 as a subsequence.
Cf. A385228 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), this sequence (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

with(numtheory):A:=3:p:=3: for c to 500 do p:=nextprime(p);if order(-2,p) mod 2=1 then A:=A,p;;fi;od:A;

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-2, #]] &] (* Paolo Xausa, Jun 30 2025 *)

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2, print1(p, ", "))); \\ Jinyuan Wang, Mar 23 2020

A139686 Odd multiplicative orders of 2 modulo primes.

Original entry on oeis.org

3, 11, 5, 23, 35, 9, 39, 11, 51, 7, 15, 83, 95, 99, 37, 29, 119, 131, 135, 155, 21, 179, 183, 191, 43, 73, 231, 239, 243, 251, 299, 25, 303, 45, 323, 359, 121, 371, 375, 411, 419, 431, 55, 443, 91, 153, 117, 483, 491, 495, 515, 519, 531, 543, 29, 575, 611, 615, 639

Offset: 1

Max Alekseyev, Apr 29 2008

nonn

Subsequence of A014664, consisting of odd elements.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A014664, A014663 (corresponding primes).

Cf. other bases: this sequence (base 2), A385226 (base 3), A385227 (base 4), A385193 (base 5), A385228 (base -2), A385229 (base -3), A385230 (base -4), A385231 (base -5).

p = Select[Range[1000], PrimeQ]; Select[MultiplicativeOrder[2, #] & /@ p, OddQ] (* Amiram Eldar, Jul 30 2020 *)

forprime(p=3,10^5,z=znorder(Mod(2,p));if(z%2,print1(z,", ")))

a(n) = multiplicative order of 2 modulo A014663(n).

A385193 Odd multiplicative orders of 5 modulo primes.

Original entry on oeis.org

1, 5, 9, 3, 29, 5, 39, 25, 27, 65, 69, 37, 75, 89, 15, 19, 33, 35, 119, 25, 67, 27, 155, 165, 179, 21, 97, 25, 17, 209, 215, 219, 115, 239, 245, 249, 135, 71, 285, 299, 309, 35, 329, 115, 359, 123, 375, 405, 9, 419, 429, 455, 459, 235, 485, 495, 509, 255, 515, 173, 525, 265, 267, 109, 575, 45

Offset: 1

Jianing Song, Jun 20 2025

a(n) is the multiplicative order of 5 modulo A385192(n).

Odd elements in A211241.

			a(8) = 25 since it is the multiplicative order of 5 modulo A385192(8) = 101, and it is odd.

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

Cf. A211241, A385192 (corresponding primes).

Cf. other bases: A139686 (base 2), A385226 (base 3), A385227 (base 4), this sequence (base 5), A385228 (base -2), A385229 (base -3), A385230 (base -4), A385231 (base -5).

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

forprime(p=2, 1e4, if(p!=5, z=znorder(Mod(5, p)); if(z%2, print1(z, ", "))))

A385226 Odd multiplicative orders of 3 modulo primes.

Original entry on oeis.org

1, 5, 3, 11, 23, 29, 35, 41, 53, 27, 65, 83, 89, 45, 95, 113, 57, 119, 125, 131, 69, 155, 39, 173, 179, 191, 209, 105, 43, 27, 221, 233, 239, 49, 251, 135, 281, 293, 299, 75, 323, 329, 31, 177, 359, 183, 371, 9, 413, 207, 419, 431, 443, 455, 473, 485, 491

Offset: 1

Jianing Song, Jun 22 2025

a(n) is the multiplicative order of 3 modulo A385220(n).

Odd elements in A062117.

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

Cf. A062117, A385220 (corresponding primes).

Cf. other bases: A139686 (base 2), this sequence (base 3), A385227 (base 4), A385193 (base 5), A385228 (base -2), A385229 (base -3), A385230 (base -4), A385231 (base -5).

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

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

A385227 Odd multiplicative orders of 4 modulo primes.

Original entry on oeis.org

1, 3, 5, 9, 11, 5, 7, 23, 29, 33, 35, 9, 39, 41, 11, 51, 53, 7, 65, 69, 15, 81, 83, 89, 95, 99, 105, 37, 113, 29, 119, 25, 131, 135, 35, 47, 51, 155, 15, 21, 173, 179, 183, 189, 191, 209, 43, 73, 221, 231, 233, 239, 243, 245, 83, 251, 261, 273, 281, 57, 293

Offset: 1

Jianing Song, Jun 22 2025

a(n) is the multiplicative order of 4 modulo A385221(n).

Odd elements in A082654.

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

Cf. A082654, A385221 (corresponding primes).

Cf. other bases: A139686 (base 2), A385226 (base 3), this sequence (base 4), A385193 (base 5), A385228 (base -2), A385229 (base -3), A385230 (base -4), A385231 (base -5).

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

forprime(p=3, 1e4, z=znorder(Mod(4, p)); if(z%2, print1(z, ", ")))

A385229 Odd multiplicative orders of -3 modulo primes.

Original entry on oeis.org

1, 3, 9, 15, 9, 21, 5, 11, 39, 17, 63, 69, 25, 39, 81, 99, 105, 111, 15, 141, 17, 165, 87, 61, 93, 189, 99, 73, 231, 243, 83, 29, 7, 285, 303, 51, 103, 315, 107, 11, 345, 121, 369, 375, 131, 405, 411, 71, 429, 219, 63, 453, 153, 117, 161, 165, 83, 17, 519, 105, 531, 267, 543, 561, 117

Offset: 1

Jianing Song, Jun 22 2025

Odd elements in A380482.

a(n) is the multiplicative order of -3 modulo A385223(n).

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

Cf. A380482, A385223 (corresponding primes).

Cf. other bases: A139686 (base 2), A385226 (base 3), A385227 (base 4), A385193 (base 5), A385228 (base -2), this sequence (base -3), A385230 (base -4), A385231 (base -5).

Select[Map[MultiplicativeOrder[-3, #] &, Prime[Range[200]]], OddQ] (* Paolo Xausa, Jun 30 2025 *)

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

A385230 Odd multiplicative orders of -4 modulo primes.

Original entry on oeis.org

1, 3, 7, 9, 5, 13, 15, 25, 9, 7, 17, 37, 13, 43, 45, 49, 19, 67, 23, 73, 39, 79, 87, 93, 97, 11, 51, 105, 19, 115, 127, 65, 135, 139, 71, 37, 153, 163, 165, 169, 175, 177, 61, 189, 95, 193, 199, 101, 205, 207, 213, 107, 219, 235, 17, 83, 23, 85, 265, 89, 91, 277, 279, 141, 59, 75

Offset: 1

Jianing Song, Jun 22 2025

Odd elements in A380531.

a(n) is the multiplicative order of -4 modulo A385224(n).

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

Cf. A380531, A385224 (corresponding primes).

Cf. other bases: A139686 (base 2), A385226 (base 3), A385227 (base 4), A385193 (base 5), A385228 (base -2), A385229 (base -3), this sequence (base -4), A385231 (base -5).

Select[Map[MultiplicativeOrder[-4, #] &, Prime[Range[200]]], OddQ] (* Paolo Xausa, Jun 30 2025 *)

forprime(p=3, 1e4, z=znorder(Mod(-4, p)); if(z%2, print1(z, ", ")))

A385231 Odd multiplicative orders of -5 modulo primes.

Original entry on oeis.org

1, 1, 3, 11, 7, 21, 23, 15, 11, 41, 51, 53, 21, 27, 83, 111, 113, 57, 131, 141, 153, 173, 87, 61, 191, 105, 221, 7, 231, 233, 27, 251, 127, 5, 261, 273, 281, 293, 303, 107, 323, 165, 341, 175, 177, 363, 371, 19, 393, 205, 137, 59, 431, 63, 443, 453, 473, 483, 491, 177, 181, 551, 277, 187, 141

Offset: 1

Jianing Song, Jun 22 2025

Odd elements in A380532.

a(n) is the multiplicative order of -5 modulo A385225(n).

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

Cf. A380532, A385225 (corresponding primes).

Cf. other bases: A139686 (base 2), A385226 (base 3), A385227 (base 4), A385193 (base 5), A385228 (base -2), A385229 (base -3), A385230 (base -4), this sequence (base -5).

Select[Map[MultiplicativeOrder[-5, #] &, Prime[Range[200]]], OddQ] (* Paolo Xausa, Jun 30 2025 *)

forprime(p=2, 1e4, if(p!=5, z=znorder(Mod(-5, p)); if(z%2, print1(z, ", "))))

cp's OEIS Frontend

A163183 Primes dividing 2^j + 1 for some odd j.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A139686 Odd multiplicative orders of 2 modulo primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385193 Odd multiplicative orders of 5 modulo primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A385226 Odd multiplicative orders of 3 modulo primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A385227 Odd multiplicative orders of 4 modulo primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A385229 Odd multiplicative orders of -3 modulo primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A385230 Odd multiplicative orders of -4 modulo primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A385231 Odd multiplicative orders of -5 modulo primes.

Views

Author

Keywords

Comments