A211242 -id:A211242 - 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)))

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.

A320383 Multiplicative order of 3/2 modulo n-th prime.

Original entry on oeis.org

2, 6, 10, 4, 16, 3, 11, 7, 30, 36, 40, 21, 23, 13, 58, 12, 33, 7, 36, 26, 82, 88, 8, 25, 102, 106, 108, 112, 126, 130, 136, 69, 74, 150, 156, 81, 83, 86, 178, 36, 95, 96, 49, 66, 5, 222, 226, 228, 232, 119, 30, 250, 256, 131, 67, 270, 276, 40, 141, 73, 51, 155, 156, 79, 11, 168, 346, 348, 352, 179, 366, 124

Offset: 3

Jianing Song, Oct 12 2018

Let p = prime(n). a(n) is the smallest positive k such that p divides 3^k - 2^k. Obviously, a(n) divides p - 1. If a(n) = p - 1, then p is listed in A320384.
If p == 1, 5, 19, 23 (mod 24), then 3/2 is a quadratic residue modulo p, so a(n) divides (p - 1)/2.
By Zsigmondy's theorem, for each k >=2 there is a prime that divides 3^k-2^k but not 3^j-2^j for j < k.  Therefore each integer >= 2 appears in the sequence at least once. - Robert Israel, Apr 20 2021

			Let ord(n,p) be the multiplicative order of n modulo p.
3/2 == 4 (mod 5), so a(3) = ord(4,5) = 2.
3/2 == 5 (mod 7), so a(4) = ord(5,7) = 6.
3/2 == 7 (mod 11), so a(5) = ord(7,11) = 10.
3/2 == 8 (mod 13), so a(6) = ord(8,13) = 4.

Robert Israel, Table of n, a(n) for n = 3..10000
Wikipedia, Multiplicative order
Wikipedia, Zsigmondy's theorem

Cf. A001047, A211242, A320384.

f:= proc(n) local p; p:= ithprime(n); numtheory:-order(3/2 mod p,p) end proc:
map(f, [$3..100]); # Robert Israel, Apr 20 2021

a[n_] := With[{p = Prime[n]}, Do[If[Divisible[3^k - 2^k, p], Return[k]], {k, Rest@Divisors[p-1]}]];
Table[a[n], {n, 3, 100}] (* Jean-François Alcover, Feb 10 2023 *)

forprime(p=5,10^3,print1(znorder(Mod(3/2,p)),", ")) \\ Joerg Arndt, Oct 13 2018

A240661 Least k such that 6^k == -1 (mod prime(n)), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 1, 5, 6, 8, 0, 0, 7, 3, 2, 20, 0, 0, 13, 29, 30, 0, 0, 18, 39, 41, 44, 6, 5, 51, 53, 54, 56, 63, 65, 68, 0, 0, 75, 78, 0, 0, 0, 89, 30, 0, 48, 7, 99, 0, 111, 113, 114, 116, 0, 10, 125, 128, 0, 67, 135, 138, 28, 0, 73, 0, 0, 26, 79, 0, 28, 173, 58, 16

Offset: 1

T. D. Noe, Apr 14 2014

nonn

The least k, if it exists, such that prime(n) divides 6^k + 1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A211242 (order of 6 mod prime(n)).

Table[p = Prime[n]; s = Select[Range[p/2], PowerMod[6, #, p] == p - 1 &, 1]; If[s == {}, 0, s[[1]]], {n, 100}]

a(n) = A211242(n)/2 if A211242(n) is even, otherwise 0.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A320383 Multiplicative order of 3/2 modulo n-th prime.

Views

Author

Keywords