A211241 -id:A211241 - cp's OEIS frontend

A380482 a(n) is the multiplicative order of -3 modulo prime(n); a(2) = 0 for completion.

1, 0, 4, 3, 10, 6, 16, 9, 22, 28, 15, 9, 8, 21, 46, 52, 58, 5, 11, 70, 12, 39, 82, 88, 48, 100, 17, 106, 54, 112, 63, 130, 136, 69, 148, 25, 39, 81, 166, 172, 178, 90, 190, 16, 196, 99, 105, 111, 226, 114, 232, 238, 120, 250, 256, 262, 268, 15, 138, 280

Offset: 1

Jianing Song, Jun 27 2025

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

Cf. A105875 (primes having primitive root -3).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), this sequence, A380531, A380532, A380533, A380540, A380541, A380542, A385222.

A380482[n_] := If[n == 2, 0, MultiplicativeOrder[-3, Prime[n]]];
Array[A380482, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-3}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

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

Original entry on oeis.org

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)))

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, ", "))))

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)))

A064132 Number of divisors of 5^n + 1 that are relatively prime to 5^m + 1 for all 0 < m < n.

Original entry on oeis.org

2, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 4, 2, 4, 4, 8, 2, 4, 4, 8, 8, 4, 16, 4, 8, 8, 4, 4, 4, 16, 4, 16, 2, 2, 2, 8, 4, 8, 8, 16, 8, 8, 2, 2, 16, 4, 2, 16, 2, 16, 4, 16, 8, 8, 4, 2, 32, 8, 4, 8, 4, 8, 8, 16, 8, 4, 16, 16, 8, 8, 16, 8, 8, 16, 8, 8, 16, 8, 8, 4, 4, 8, 16, 8, 8, 32, 16, 2, 16

Offset: 0

Robert G. Wilson v, Sep 10 2001

nonn

From Robert Israel, Jun 26 2018: (Start)
a(n) = Product_{j: A211241(j)=2*n} (1 + e_j) where e_j is the Prime(j)-adic valuation of 5^n+1.  In most cases, each e_j = 1 and a(n) is a power of 2, but a(20243) is divisible by 3 since the multiplicative order of 5 mod 40487 is 40486 and 5^20243+1 is divisible by 40487^2.
(End)

Sam Wagstaff, Cunningham Project, Factorizations of 5^n-1, n odd, n<=375

Cf. A064131, A064133, A064134, A064135, A064136, A064137.

Cf. A211241.

f:= n -> nops(select(t -> andmap(m -> igcd(t,5^m+1)=1,[$1..n-1]), numtheory:-divisors(5^n+1))):
map(f, [$0..100]); # Robert Israel, Jun 25 2018

a[n_] := Count[Divisors[5^n+1], d_ /; AllTrue[5^Range[n-1]+1, CoprimeQ[d, #]&]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 100}] (* Jean-François Alcover, Jun 27 2018 *)

a(n) = if (n==0, 2, sumdiv(5^n+1, d, vecsum(vector(n-1, k, gcd(d, 5^k+1) == 1)) == n-1)); \\ Michel Marcus, Jun 24 2018

More terms from Robert Israel, Jun 25 2018

Incorrect Mma program deleted by Editors, Jul 02 2018

cp's OEIS Frontend

A380482 a(n) is the multiplicative order of -3 modulo prime(n); a(2) = 0 for completion.

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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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A385193 Odd multiplicative orders of 5 modulo primes.

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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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