A014663 -id:A014663 - cp's OEIS frontend

A261580 Primes p such that A214028(p) is odd.

5, 13, 29, 37, 53, 61, 101, 109, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 317, 349, 373, 389, 397, 421, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 773, 797, 821, 829, 853, 857, 877, 941, 953, 997, 1013, 1021, 1061, 1069

Offset: 1

Michel Marcus, Aug 25 2015

nonn

From Jianing Song, Aug 13 2019: (Start)
Primes p with 4 zeros in a fundamental period of A000129 mod p, that is, primes p such that A214027(p) = 4. For a proof of the equivalence between A214027(p) = 4 and A214028(p) being odd, see Section 2 of my link below.
For p > 2, p is in this sequence if and only if A175181(p) == 4 (mod 8).
This sequence contains all primes congruent to 5 modulo 8. This corresponds to case (1) for k = 6 in the Conclusion of Section 1 of my link below.
Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 7/24 in the primes; see the end of Section 1 of my link. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 20 2024]
The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

			The smallest Pell number divisible by the prime 5 has index 3, which is odd, so 5 is in the sequence.

Jianing Song, Table of n, a(n) for n = 1..1280
Bernadette Faye and Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.
Jianing Song, Lucas sequences and entry point modulo p

Cf. A214028, A261581.
Cf. also A175181.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | A309580  | A309586
Primes p such that w(p) = 2  | A053027  | A309581  | A309587
Primes p such that w(p) = 4  | A053028  | this seq | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
* and also A053032 U {2}

f[n_] := Block[{k = 1}, While[Mod[Simplify[((1 + Sqrt@ 2)^k - (1 - Sqrt@ 2)^k)/(2 Sqrt@ 2)], n] != 0, k++]; k]; Select[Prime@ Range@ 180, OddQ@ f@ # &] (* Michael De Vlieger, Aug 25 2015 *)

pell(n) = polcoeff(Vec(x/(1-2*x-x^2) + O(x^(n+1))), n);
z(n) = {k=1; while (pell(k) % n, k++); k;}
lista(nn) = {forprime(p=2, nn, if (z(p) % 2, print1(p, ", ")););}

forprime(p=2, 1100, if(A214027(p)==4, print1(p, ", "))) \\ Jianing Song, Aug 13 2019

A309580 Primes p with 1 zero in a fundamental period of A000129 mod p.

Original entry on oeis.org

2, 7, 23, 31, 41, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 313, 353, 359, 367, 383, 409, 431, 439, 457, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 761, 809, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1129, 1151, 1201, 1223, 1231, 1279

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A214027(p) = 1.
For p > 2, p is in this sequence if and only if A175181(p) == 2 (mod 4), and if and only if A214028(p) == 2 (mod 4). For a proof of the equivalence between A214027(p) = 1 and A214028(p) == 2 (mod 4), see Section 2 of my link below.
This sequence contains all primes congruent to 7 modulo 8. This corresponds to case (3) for k = 6 in the Conclusion of Section 1 of my link below.
Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 7/24 in the primes; see the end of Section 1 of my link. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]
The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

Jianing Song, Table of n, a(n) for n = 1..1280
Jianing Song, Lucas sequences and entry point modulo p

Cf. A175181, A214028.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | this seq | A309586
Primes p such that w(p) = 2  | A053027  | A309581  | A309587
Primes p such that w(p) = 4  | A053028  | A261580  | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
* and also A053032 U {2}

forprime(p=2, 1300, if(A214027(p)==1, print1(p, ", ")))

A309581 Primes p with 2 zeros in a fundamental period of A000129 mod p.

Original entry on oeis.org

3, 11, 17, 19, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 331, 337, 347, 379, 401, 419, 433, 443, 449, 467, 491, 499, 523, 547, 563, 571, 577, 587, 601, 617, 619, 641, 643, 659, 673, 683, 691

Offset: 1

Jianing Song, Aug 10 2019

nonn

Primes p such that A214027(p) = 2.
For p > 2, p is in this sequence if and only if 8 divides A175181(p), and if and only if 4 divides A214028(p). For a proof of the equivalence between A214027(p) = 2 and 4 dividing A214028(p), see Section 2 of my link below.
This sequence contains all primes congruent to 3 modulo 8. This corresponds to case (2) for k = 6 in the Conclusion of Section 1 of my link below.
Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 5/12 in the primes; see the end of Section 1 of my link. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]
The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

Jianing Song, Table of n, a(n) for n = 1..1600
Jianing Song, Lucas sequences and entry point modulo p

Cf. A175181, A214028.
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
                             |   m=1    |   m=2    |   m=3
-----------------------------+----------+----------+---------
The sequence {x(n)}          | A000045  | A000129  | A006190
The sequence {w(k)}          | A001176  | A214027  | A322906
Primes p such that w(p) = 1  | A112860* | A309580  | A309586
Primes p such that w(p) = 2  | A053027  | this seq | A309587
Primes p such that w(p) = 4  | A053028  | A261580  | A309588
Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
* and also A053032 U {2}

forprime(p=2, 700, if(A214027(p)==2, print1(p, ", ")))

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

Original entry on oeis.org

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

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

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).

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

cp's OEIS Frontend

A261580 Primes p such that A214028(p) is odd.

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A309580 Primes p with 1 zero in a fundamental period of A000129 mod p.

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A309581 Primes p with 2 zeros in a fundamental period of A000129 mod p.

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

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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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A139686 Odd multiplicative orders of 2 modulo primes.

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

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