A082654 -id:A082654 - cp's OEIS frontend

A380531 a(n) is the multiplicative order of -4 modulo prime(n); a(1) = 0 for completion.

0, 2, 1, 6, 10, 3, 4, 18, 22, 7, 10, 9, 5, 14, 46, 13, 58, 15, 66, 70, 18, 78, 82, 22, 24, 25, 102, 106, 9, 7, 14, 130, 17, 138, 37, 30, 13, 162, 166, 43, 178, 45, 190, 48, 49, 198, 210, 74, 226, 19, 58, 238, 12, 50, 8, 262, 67, 270, 23, 70

Offset: 1

Jianing Song, Jun 27 2025

a(n) divides (p-1)/4 if p = prime(n) == 1 (mod 4), since (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i^2 == -1 (mod p).

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

Cf. A105876 (primes having primitive root -4).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, this sequence, A380532, A380533, A380540, A380541, A380542, A385222.

A380531[n_] := If[n == 1, 0, MultiplicativeOrder[-4, Prime[n]]];
Array[A380531, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-4}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380532 a(n) is the multiplicative order of -5 modulo prime(n); a(3) = 0 for completion.

Original entry on oeis.org

1, 1, 0, 3, 10, 4, 16, 18, 11, 7, 6, 36, 20, 21, 23, 52, 58, 15, 11, 10, 72, 78, 41, 44, 96, 50, 51, 53, 54, 112, 21, 130, 136, 138, 74, 150, 156, 27, 83, 172, 178, 30, 38, 192, 196, 66, 70, 111, 113, 57, 232, 238, 40, 50, 256, 131, 134, 54, 276, 140

Offset: 1

Jianing Song, Jun 27 2025

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

Cf. A105877 (primes having primitive root -5).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, this sequence, A380533, A380540, A380541, A380542, A385222.

A380532[n_] := If[n == 3, 0, MultiplicativeOrder[-5, Prime[n]]];
Array[A380532, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-5}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380533 a(n) is the multiplicative order of -6 modulo prime(n); a(1) = a(2) = 0 for completion.

Original entry on oeis.org

0, 0, 2, 1, 5, 12, 16, 18, 22, 7, 3, 4, 40, 6, 46, 13, 29, 60, 66, 70, 36, 39, 41, 88, 12, 5, 51, 53, 108, 112, 63, 65, 136, 46, 74, 75, 156, 54, 166, 86, 89, 60, 38, 96, 7, 99, 210, 111, 113, 228, 232, 34, 20, 125, 256, 262, 67, 135, 276, 56

Offset: 1

Jianing Song, Jun 27 2025

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

Cf. A105878 (primes having primitive root -6).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, this sequence, A380540, A380541, A380542, A385222.

A380533[n_] := If[n < 3, 0, MultiplicativeOrder[-6, Prime[n]]];
Array[A380533, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-6}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380540 a(n) is the multiplicative order of -7 modulo prime(n); a(4) = 0 for completion.

Original entry on oeis.org

1, 2, 4, 0, 5, 12, 16, 6, 11, 14, 30, 18, 40, 3, 46, 13, 58, 60, 33, 35, 24, 39, 82, 88, 96, 100, 102, 53, 54, 7, 63, 130, 68, 138, 37, 75, 52, 81, 166, 172, 89, 12, 5, 24, 49, 198, 105, 74, 226, 228, 116, 119, 240, 250, 256, 131, 268, 270, 69, 20

Offset: 1

Jianing Song, Jun 27 2025

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

Cf. A105879 (primes having primitive root -7).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, this sequence, A380541, A380542, A385222.

A380540[n_] := If[n == 4, 0, MultiplicativeOrder[-7, Prime[n]]];
Array[A380540, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-7}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380541 a(n) is the multiplicative order of -8 modulo prime(n); a(1) = 0 for completion.

Original entry on oeis.org

0, 1, 4, 2, 5, 4, 8, 3, 22, 28, 10, 12, 20, 7, 46, 52, 29, 20, 11, 70, 6, 26, 41, 22, 16, 100, 34, 53, 12, 28, 14, 65, 68, 23, 148, 10, 52, 27, 166, 172, 89, 60, 190, 32, 196, 66, 35, 74, 113, 76, 58, 238, 8, 25, 16, 262, 268, 90, 92, 35

Offset: 1

Jianing Song, Jun 27 2025

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

Cf. A105880 (primes having primitive root -8).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, A380540, this sequence, A380542, A385222.

A380541[n_] := If[n == 1, 0, MultiplicativeOrder[-8, Prime[n]]];
Array[A380541, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-8}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

a(n) = ord(-2,p)/gcd(ord(-2,p),3) for p != 2, where p = prime(n), and ord(a,m) is the multiplicative order of a modulo m. Note that ord(-2,p) = A337878(n) for n > 2.

A380542 a(n) is the multiplicative order of -9 modulo prime(n); a(2) = 0 for completion.

Original entry on oeis.org

1, 0, 1, 6, 10, 6, 8, 18, 22, 7, 30, 18, 4, 42, 46, 13, 58, 10, 22, 70, 3, 78, 82, 44, 24, 25, 34, 106, 54, 56, 126, 130, 68, 138, 37, 50, 78, 162, 166, 43, 178, 90, 190, 8, 49, 198, 210, 222, 226, 114, 116, 238, 60, 250, 128, 262, 67, 30, 138, 140

Offset: 1

Jianing Song, Jun 27 2025

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

Cf. A105881 (primes having primitive root -9).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, A380540, A380541, this sequence, A385222.

A380542[n_] := If[n == 2, 0, MultiplicativeOrder[-9, Prime[n]]];
Array[A380542, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-9}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A385222 a(n) is the multiplicative order of -10 modulo prime(n); a(1) = a(3) = 0 for completion.

Original entry on oeis.org

0, 2, 0, 3, 1, 3, 16, 9, 11, 28, 30, 6, 10, 42, 23, 26, 29, 60, 66, 70, 8, 26, 82, 44, 96, 4, 17, 106, 108, 112, 21, 65, 8, 23, 148, 150, 39, 162, 83, 86, 89, 180, 190, 192, 49, 198, 15, 111, 226, 228, 232, 14, 15, 25, 256, 131, 268, 10, 138, 28

Offset: 1

Jianing Song, Jun 27 2025

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

Cf. A007348 (primes having primitive root -10).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, A380540, A380541, A380542, this sequence.

A385222[n_] := If[n == 1 || n == 3, 0, MultiplicativeOrder[-10, Prime[n]]];
Array[A385222, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-10}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

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, ", ")))

A372801 Order of 16 modulo the n-th prime: least k such that prime(n) divides 16^k-1.

Original entry on oeis.org

1, 1, 3, 5, 3, 2, 9, 11, 7, 5, 9, 5, 7, 23, 13, 29, 15, 33, 35, 9, 39, 41, 11, 12, 25, 51, 53, 9, 7, 7, 65, 17, 69, 37, 15, 13, 81, 83, 43, 89, 45, 95, 24, 49, 99, 105, 37, 113, 19, 29, 119, 6, 25, 4, 131, 67, 135, 23, 35, 47, 73, 51, 155, 39, 79, 15, 21, 173, 87, 22, 179

Offset: 2

Jianing Song, May 13 2024

a(n) is the period of the expansion of 1/prime(n) in hexadecimal.

Jianing Song, Table of n, a(n) for n = 2..10000

Cf. A302141 (order of 16 mod 2n+1).

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.

PARI
```
a(n) = znorder(Mod(16, prime(n))).
```

a(n) = A014664(n)/gcd(4, A014664(n)) = A082654(n)/gcd(2, A082654(n)).

a(n) <= (prime(n) - 1)/2.

A372797 Smallest prime p such that the multiplicative order of 4 modulo p is 2*n, or 0 if no such prime exists.

Original entry on oeis.org

3, 17, 31, 73, 151, 433, 631, 337, 127, 241, 331, 601, 4421, 673, 3061, 257, 1429, 1657, 1103, 3121, 2143, 1321, 18539, 1777, 2351, 37441, 2971, 2857, 3191, 17401, 683, 15809, 17029, 9929, 38431, 1801, 11471, 63689, 49999, 13121, 17467, 21169, 83077, 25609, 5581, 5153, 26227

Offset: 1

Jianing Song, May 13 2024

nonn

First prime p such that the expansion of 1/p has period (p-1)/(2*n) in base 4. Also the first prime p such that {k/p : 1 <= k <= p-1} has 2*n different cycles when written out in base 4.

Since ord(a^m,k) = ord(a,k)/gcd(m,ord(a,k)) for gcd(a,k) = 1, we have that (p-1)/ord(4,p) = ((p-1)/ord(2,p)) * gcd(2,ord(2,p)) is always even. Here ord(a,k) is the multiplicative order of a modulo k.

			In the following examples let () denote the reptend. The prime numbers themselves and the fractions are written out in decimal.
The base-4 expansion of 1/3 is 0.(1), so the reptend has length 1 = (3-1)/2. Also, the base-4 expansions of 1/3 = 0.(1) and 2/3 = 0.(2) have two cycles 1 and 2. 3 is the smallest such prime, so a(1) = 3.
The base-4 expansion of 1/17 is 0.(0033), so the reptend has length 4 = (17-1)/4. Also, the base-4 expansions of 1/17, 2/17, ..., 16/17 have four cycles 0033, 0132, 1023 and 1122. 17 is the smallest such prime, so a(2) = 17.
The base-4 expansion of 1/31 is 0.(00133), so the reptend has length 5 = (31-1)/6. Also, the base-4 expansions of 1/31, 2/31, ..., 30/31 have three cycles 00201, 01203, 02211, 03213, 11223 and 13233. 13 is the smallest such prime, so a(3) = 13.

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Cf. A082654, A216371.

In other bases: A101208, A101209, A372798, A372799, A054471, A372800.

a[n_] := a[n] = For[p = 2, True, p = NextPrime[p], If[MultiplicativeOrder[4, p] == (p-1)/(2n), Return[p]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 24 2024 *)

a(n,{base=4}) = forprime(p=2, oo, if((base%p) && znorder(Mod(base,p)) == (p-1)/(n * if(issquare(base), 2, 1)), return(p)))

cp's OEIS Frontend

A380531 a(n) is the multiplicative order of -4 modulo prime(n); a(1) = 0 for completion.

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A380532 a(n) is the multiplicative order of -5 modulo prime(n); a(3) = 0 for completion.

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A380533 a(n) is the multiplicative order of -6 modulo prime(n); a(1) = a(2) = 0 for completion.

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A380540 a(n) is the multiplicative order of -7 modulo prime(n); a(4) = 0 for completion.

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A380541 a(n) is the multiplicative order of -8 modulo prime(n); a(1) = 0 for completion.

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A380542 a(n) is the multiplicative order of -9 modulo prime(n); a(2) = 0 for completion.

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A385222 a(n) is the multiplicative order of -10 modulo prime(n); a(1) = a(3) = 0 for completion.

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A385227 Odd multiplicative orders of 4 modulo primes.

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A372801 Order of 16 modulo the n-th prime: least k such that prime(n) divides 16^k-1.

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