cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Max Alekseyev, Apr 29 2008

Keywords

Comments

Subsequence of A014664, consisting of odd elements.

Links

Crossrefs

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).

Programs

  • Mathematica
    p = Select[Range[1000], PrimeQ]; Select[MultiplicativeOrder[2, #] & /@ p, OddQ] (* Amiram Eldar, Jul 30 2020 *)
  • PARI
    forprime(p=3,10^5,z=znorder(Mod(2,p));if(z%2,print1(z,", ")))

Formula

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

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

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

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.

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-4, #]] &] (* Paolo Xausa, Jun 28 2025 *)
  • PARI
    isA385224(p) = isprime(p) && (p!=2) && znorder(Mod(-4,p))%2

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

Views

Author

Jianing Song, Jun 20 2025

Keywords

Comments

a(n) is the multiplicative order of 5 modulo A385192(n).
Odd elements in A211241.

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

a(n) is the multiplicative order of 3 modulo A385220(n).
Odd elements in A062117.

Links

Crossrefs

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).

Programs

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

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

a(n) is the multiplicative order of 4 modulo A385221(n).
Odd elements in A082654.

Links

Crossrefs

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).

Programs

  • Mathematica
    Select[Map[MultiplicativeOrder[4, #] &, Prime[Range[200]]], OddQ] (* Paolo Xausa, Jun 28 2025 *)
  • PARI
    forprime(p=3, 1e4, z=znorder(Mod(4, p)); if(z%2, print1(z, ", ")))

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

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

a(n) is the multiplicative order of -2 modulo A163183(n).
Odd elements in A337878 (with first term changed to 1).

Links

Crossrefs

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).

Programs

  • Mathematica
    Select[Map[MultiplicativeOrder[-2, #] &, Prime[Range[250]]], OddQ] (* Paolo Xausa, Jun 28 2025 *)
  • PARI
    forprime(p=3, 1e4, z=znorder(Mod(-2, 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

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

Odd elements in A380482.
a(n) is the multiplicative order of -3 modulo A385223(n).

Links

Crossrefs

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).

Programs

  • Mathematica
    Select[Map[MultiplicativeOrder[-3, #] &, Prime[Range[200]]], OddQ] (* Paolo Xausa, Jun 30 2025 *)
  • PARI
    forprime(p=2, 1e4, if(p!=3, z=znorder(Mod(-3, 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

Views

Author

Jianing Song, Jun 22 2025

Keywords

Comments

Odd elements in A380532.
a(n) is the multiplicative order of -5 modulo A385225(n).

Links

Crossrefs

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).

Programs

  • Mathematica
    Select[Map[MultiplicativeOrder[-5, #] &, Prime[Range[200]]], OddQ] (* Paolo Xausa, Jun 30 2025 *)
  • PARI
    forprime(p=2, 1e4, if(p!=5, z=znorder(Mod(-5, p)); if(z%2, print1(z, ", "))))
Showing 1-8 of 8 results.

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