A053451 -id:A053451 - cp's OEIS frontend

A070680 Smallest m in range 1..phi(n) such that 11^m == 1 mod n, or 0 if no such number exists.

0, 1, 2, 2, 1, 2, 3, 2, 6, 1, 0, 2, 12, 3, 2, 4, 16, 6, 3, 2, 6, 0, 22, 2, 5, 12, 18, 6, 28, 2, 30, 8, 0, 16, 3, 6, 6, 3, 12, 2, 40, 6, 7, 0, 6, 22, 46, 4, 21, 5, 16, 12, 26, 18, 0, 6, 6, 28, 58, 2, 4, 30, 6, 16, 12, 0, 66, 16, 22, 3, 70, 6, 72, 6, 10, 6, 0, 12, 39, 4

Offset: 1

N. J. A. Sloane and Amarnath Murthy, May 08 2002

nonn

Cf. A070667-A070675, A002326, A070676, A053447, A070677, A070681, A070678, A053451, A070679, A070682, A070683.

[0] cat [Modorder(11, n): n in [2..100]]; // Vincenzo Librandi, Apr 01 2014

Table[SelectFirst[Range[EulerPhi[n]],PowerMod[11,#,n]==1&,0],{n,80}] (* Paul F. Marrero Romero, Oct 21 2024 *)

A070681 Smallest m in range 1..phi(2n+1) such that 6^m == 1 mod 2n+1, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 1, 2, 0, 10, 12, 0, 16, 9, 0, 11, 5, 0, 14, 6, 0, 2, 4, 0, 40, 3, 0, 23, 14, 0, 26, 10, 0, 58, 60, 0, 12, 33, 0, 35, 36, 0, 10, 78, 0, 82, 16, 0, 88, 12, 0, 9, 12, 0, 10, 102, 0, 106, 108, 0, 112, 11, 0, 16, 110, 0, 25, 126, 0, 130, 18, 0, 136, 23, 0, 60

Offset: 0

N. J. A. Sloane and Amarnath Murthy, May 08 2002

nonn

Cf. A070667-A070675, A002326, A070676, A053447, A070677, A070678, A053451, A070679, A070682, A070680, A070683.

A302141 Multiplicative order of 16 mod 2n+1.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 3, 1, 2, 9, 3, 11, 5, 9, 7, 5, 5, 3, 9, 3, 5, 7, 3, 23, 21, 2, 13, 5, 9, 29, 15, 3, 3, 33, 11, 35, 9, 5, 15, 39, 27, 41, 2, 7, 11, 3, 5, 9, 12, 15, 25, 51, 3, 53, 9, 9, 7, 11, 3, 6, 55, 5, 25, 7, 7, 65, 9, 9, 17, 69, 23, 15, 7, 21, 37, 15, 6, 5, 13, 13, 33, 81, 5, 83, 39, 9, 43, 15, 29, 89, 45, 15, 9, 10, 9, 95, 24, 3, 49, 99, 33

Offset: 0

Jianing Song, Apr 02 2018

nonn

Reptend length of 1/(2n+1) in hexadecimal.
a(n) <= n; it appears that equality holds if and only if n=1 or is in A163778. - Robert Israel, Apr 02 2018
From Jianing Song, Dec 24 2022: (Start)
a(n) <= psi(2*n+1)/2 <= n. a(n) = psi(2*n+1)/2 if and only if the multiplicative order of 2 modulo 2*n+1 is psi(2*n+1) or psi(2*n+1)/2, and psi(2*n+1) == 2 (mod 4).
a(n) = n if and only if A053447(n) = n and A053447(n) is odd. As a result, a(n) = n if and only if 2*n+1 = p is a prime congruent to 3 modulo 4, and the multiplicative order of 2 modulo p is p-1 or (p-1)/2 (p-1 if p == 3 (mod 8), (p-1)/2 if p == 7 (mod 8)). Such primes p are listed in A105876. (End)

			The fraction 1/13 is equal to 0.13B13B... in hexadecimal, so a(6) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Jianing Song)
Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A053447, A053451, A105876, A163778.

List([0..100],n->OrderMod(16,2*n+1)); # Muniru A Asiru, Feb 25 2019

[1] cat [ Modorder(16, 2*n+1): n in [1..100]]; // Vincenzo Librandi, Apr 03 2018

seq(numtheory:-order(16, 2*n+1), n=0..100); # Robert Israel, Apr 02 2018

Table[MultiplicativeOrder[16, 2 n + 1], {n, 0, 150}] (* Vincenzo Librandi, Apr 03 2018 *)

a(n) = znorder(Mod(16, 2*n+1)) \\ Felix Fröhlich, Apr 02 2018

a(n) = A002326(n)/gcd(A002326(n),4) = A053447(n)/gcd(A053447(n),2). [Corrected by Jianing Song, Dec 24 2022]

A050980 Haupt-exponents of 8 modulo integers relatively prime to 8.

Original entry on oeis.org

2, 4, 1, 2, 10, 4, 4, 8, 6, 2, 11, 20, 6, 28, 5, 10, 4, 12, 4, 20, 14, 4, 23, 7, 8, 52, 20, 6, 58, 20, 2, 4, 22, 22, 35, 3, 20, 10, 13, 18, 82, 8, 28, 11, 4, 10, 12, 16, 10, 100, 17, 4, 106, 12, 12, 28, 44, 4, 8, 110, 20, 100, 7, 14, 130, 6, 12, 68, 46, 46, 20, 28, 14, 148, 5

Offset: 1

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A002329. Presumably this is a duplicate of A053451.

A266697 Multiplicative order of 2^n mod 2*n+1.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 2, 4, 1, 2, 3, 1, 5, 18, 2, 1, 5, 12, 2, 12, 1, 2, 6, 1, 7, 8, 2, 20, 9, 2, 2, 6, 3, 2, 11, 1, 1, 20, 15, 1, 27, 2, 4, 28, 1, 4, 5, 36, 1, 30, 2, 1, 3, 2, 2, 36, 1, 44, 6, 24, 11, 20, 50, 1, 7, 2, 3, 36, 1, 2, 23, 60, 7, 42, 2, 1, 6, 20, 2

Offset: 0

Vincenzo Librandi, Jan 03 2016

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A053447, A053451.

List([0..100],n->OrderMod(2^n,2*n+1)); # Muniru A Asiru, Feb 25 2019

[1] cat [Modorder(2^n, 2*n+1): n in [1..100]];

1,seq(numtheory:-order(2^n,2*n+1),n=1..100); # Robert Israel, Jan 10 2016

Table[MultiplicativeOrder[2^n, 2 n + 1], {n, 0, 100}]

a(n) = if(n<0, 0, znorder(Mod(2^n, 2*n+1))); \\ Altug Alkan, Jan 04 2016

a(n) = A002326(n)/gcd(n,A002326(n)). - Robert Israel, Jan 10 2016

cp's OEIS Frontend

A070680 Smallest m in range 1..phi(n) such that 11^m == 1 mod n, or 0 if no such number exists.

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A070681 Smallest m in range 1..phi(2n+1) such that 6^m == 1 mod 2n+1, or 0 if no such number exists.

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A302141 Multiplicative order of 16 mod 2n+1.

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A050980 Haupt-exponents of 8 modulo integers relatively prime to 8.

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A266697 Multiplicative order of 2^n mod 2*n+1.

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