A053447 -id:A053447 - cp's OEIS frontend

A070678 Smallest m in range 1..phi(n) such that 7^m == 1 mod n, or 0 if no such number exists.

0, 1, 1, 2, 4, 1, 0, 2, 3, 4, 10, 2, 12, 0, 4, 2, 16, 3, 3, 4, 0, 10, 22, 2, 4, 12, 9, 0, 7, 4, 15, 4, 10, 16, 0, 6, 9, 3, 12, 4, 40, 0, 6, 10, 12, 22, 23, 2, 0, 4, 16, 12, 26, 9, 20, 0, 3, 7, 29, 4, 60, 15, 0, 8, 12, 10, 66, 16, 22, 0, 70, 6, 24, 9, 4, 6, 0, 12

Offset: 1

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

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A070667-A070675, A002326, A070676, A053447, A070677, A070681, A053451, A070679, A070682, A070680, A070683.

[0] cat [Modorder(7, n): n in [2..100]]; // Vincenzo Librandi, Apr 01 2014

Table[SelectFirst[Range[EulerPhi[n]],PowerMod[7,#,n]==1&],{n,80}]/.(Missing["NotFound"]->0) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 25 2019 *)

A070679 Smallest m in range 1..phi(n) such that 9^m == 1 mod n, or 0 if no such number exists.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 3, 1, 0, 2, 5, 0, 3, 3, 0, 2, 8, 0, 9, 2, 0, 5, 11, 0, 10, 3, 0, 3, 14, 0, 15, 4, 0, 8, 6, 0, 9, 9, 0, 2, 4, 0, 21, 5, 0, 11, 23, 0, 21, 10, 0, 3, 26, 0, 10, 3, 0, 14, 29, 0, 5, 15, 0, 8, 6, 0, 11, 8, 0, 6, 35, 0, 6, 9, 0, 9, 15, 0, 39, 2, 0, 4, 41, 0

Offset: 1

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

nonn

Cf. A070667-A070675, A002326, A070676, A053447, A070677, A070681, A070678, A053451, A070682, A070680, A070683.

[0] cat [Modorder(9, n): n in [2..100]]; // Vincenzo Librandi, Apr 01 2014

Table[SelectFirst[Range[EulerPhi[n]],PowerMod[9,#,n]==1&],{n,90}]/. Missing[ "NotFound"] -> 0 (* Harvey P. Dale, Jan 22 2023 *)

a(n) = {for (i = 1, eulerphi(n), if ((9^i % n) == 1, return(i));); return (0);} \\Michel Marcus, Jul 31 2013

A070676 Smallest m in range 1..phi(n) such that 3^m == 1 mod n, or 0 if no such number exists.

Original entry on oeis.org

0, 1, 0, 2, 4, 0, 6, 2, 0, 4, 5, 0, 3, 6, 0, 4, 16, 0, 18, 4, 0, 5, 11, 0, 20, 3, 0, 6, 28, 0, 30, 8, 0, 16, 12, 0, 18, 18, 0, 4, 8, 0, 42, 10, 0, 11, 23, 0, 42, 20, 0, 6, 52, 0, 20, 6, 0, 28, 29, 0, 10, 30, 0, 16, 12, 0, 22, 16, 0, 12, 35, 0, 12, 18, 0, 18, 30, 0

Offset: 1

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

nonn

Cf. A070667-A070675, A002326, A053447, A070677, A070681, A070678, A053451, A070679, A070682, A070680, A070683.

[0] cat [Modorder(3, n): n in [2..100]]; // Vincenzo Librandi, Apr 01 2014

Table[SelectFirst[Range[EulerPhi[n]],PowerMod[3,# ,n]==1&,0],{n,80}] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Aug 18 2015 *)

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

Original entry on oeis.org

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.

A003573 Order of 4 mod 4n+1.

Original entry on oeis.org

1, 2, 3, 6, 4, 3, 10, 14, 5, 18, 10, 6, 21, 26, 9, 30, 6, 11, 9, 15, 27, 4, 11, 5, 24, 50, 6, 18, 14, 6, 55, 50, 7, 9, 34, 23, 14, 74, 12, 26, 33, 10, 78, 86, 29, 90, 18, 9, 48, 98, 33, 10, 45, 35, 15, 12, 30, 38, 29, 39, 12, 42, 41, 55, 8, 42, 26, 134, 6, 46, 35

Offset: 0

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..10000

Cf. A003574. First bisection of A053447.

Cf. A002326, A003571, A217469.

List([0..70],n->OrderMod(4,4*n+1)); # Muniru A Asiru, Feb 16 2019

a := n -> `if`(n=0, 1, numtheory:-order(4, 4*n+1)): seq(a(n), n = 0..68);

Table[MultiplicativeOrder[4, 4*n + 1], {n, 0, 70}] (* Arkadiusz Wesolowski, Nov 27 2012 *)

a(n) = znorder(Mod(4, 4*n+1)); \\ Michel Marcus, Feb 16 2019

def A003573(n):
    s, m, N = 0, 1, 4*n + 1
    while True:
        k = N + m
        v = valuation(k, 4)
        s += v
        m = k // 4^v
        if m == 1: break
    return s
print([A003573(n) for n in (0..70)]) # Peter Luschny, Oct 07 2017

a(n) = A053447(2*n) for n >= 0. - Jianing Song, Oct 03 2022

a(0) = 1 added by Peter Luschny, Oct 07 2017

A003574 Order of 4 mod 4n-1.

Original entry on oeis.org

1, 3, 5, 2, 9, 11, 9, 5, 6, 6, 7, 23, 4, 10, 29, 3, 33, 35, 10, 39, 41, 14, 6, 18, 15, 51, 53, 18, 22, 12, 10, 7, 65, 18, 69, 30, 21, 15, 10, 26, 81, 83, 9, 30, 89, 30, 20, 95, 6, 99, 42, 33, 105, 14, 9, 37, 113, 15, 46, 119

Offset: 1

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A003573. Second bisection of A053447.

Cf. A002326, A003572, A217852.

List([1..70],n->OrderMod(4,4*n-1)); # Muniru A Asiru, Feb 19 2019

with(numtheory): f := n->order(4,4*n-1);

a(n) = znorder(Mod(4, 4*n-1)); \\ Michel Marcus, Feb 22 2019

a(n) = A053447(2*n-1) for n >= 1. - Jianing Song, Oct 03 2022

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]

A339046 Irregular triangle read by rows: row n gives the complete quadrupling system modulo N = 2*n + 1, for n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 3, 1, 4, 2, 3, 5, 6, 1, 4, 7, 2, 8, 5, 1, 4, 5, 9, 3, 2, 8, 10, 7, 6, 1, 4, 3, 12, 9, 10, 2, 8, 6, 11, 5, 7, 1, 4, 2, 8, 7, 13, 11, 14, 1, 4, 16, 13, 2, 8, 15, 9, 3, 12, 14, 5, 6, 7, 11, 10, 1, 4, 16, 2, 8, 11, 5, 20, 17, 10, 19, 13, 1, 4, 16, 18, 3, 12, 2, 8, 9, 13, 6, 2, 8, 9, 13, 6, 1, 4, 16, 18, 3, 12

Offset: 0

Wolfdieter Lang, Dec 13 2020

The length of row n is given by phi(2*n + 1), with phi = A000010, for n >= 0.
The quadrupling sequence modulo N = 2*n + 1, for n >= 0, has entries QS(N, s(N,i), j) = s(N,i)*4^j (mod N), with j >= 0, and certain positive integer seeds s(N, i), for i = 1, 2, ..., S(N) = A339049((N-1)/2), where gcd(s(N, i), N) = 1 (restricted seeds modulo N). These quadrupling sequences are periodic with period length P(N) = A053447((N-1)/2) (order of 4 modulo N). Only the periods (cycles) QS(N, s(N,i)) = {QS(N, s(N, i), j)}_{j=0..P(N)-1}, for i = 1, 2, ..., S(N), are listed.
N = 1 (n = 0) is special: one takes here the restricted residue system modulo N not as [0] but as [1]. The order of 4 modulo 1 is 1, because 4^1 == 1 (mod 1) (== 0 (mod 1)).
In order to obtain the complete system of quadrupling sequences one starts with seed s(N, 1) = 1, and if all numbers from the smallest positive reduced residue system modulo N (called RRS(N), given in row N of A038566) are obtained, i.e., if P(N) = #RRS(N) = phi(N) = A000010(N), then the system is complete. Otherwise the smallest missing number from RRS(N) is taken as new seed s(N, 2), etc. until the system is complete. This means that the number of seeds needed is S(N) given above.
This entry generalizes A337712, given together with Gary W. Adamson. See also A337936.

			The irregular triangle begins (the vertical bar separates the cycles):
n,  N \ k  1 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 ...
0,  1:     1
1,  3:     1|2
2,  5:     1 4| 2  3
3,  7:     1 4  2| 3  5  6
4,  9:     1 4  7| 2  8  5
5,  11:    1 4  5  9  3| 2  8 10  7  6
6,  13:    1 4  3 12  9 10| 2  8  6 11  5  7
7,  15:    1 4| 2  8| 7 13|11 14
8,  17:    1 4 16 13| 2  8 15  9| 3 12 14  5| 6  7 11 10
9,  19:    1 4 16  7  9 17 11  6  5| 2  8 13 14 18 15  3 12 10
10, 21:    1 4 16| 2  8 11| 5 20 17|10 19 13
11, 23:    1 4 16 18  3 12  2  8  9 13  6| 2  8  9 13  6  1  4 16 18  3 12
12, 25:    1 4 16 14  6 24 21  9 11 19| 2  8  7  3 12 23 17 18 22 13
13, 27:    1 4 16 10 13 25 19 22  7| 2  8  5 20 26 23 11 17 14
...
n = 14, N = 29: 1 4 16 6 24 9 7 28 25 13 23 5 20 22 | 2 8 3 12 19 18 14 27 21 26 17 10 11 15,
n = 15, N = 31: 1 4 16 2 8 | 3 12 17 6 24 | 5 20 18 10 9 | 7 28 19 14 25 | 11 13 21 22 26 | 15 29 23 30 27.
...

Cf. A000010, A053447, A337712 (doubling), A337936 (tripling), A339049.

T(n, k) gives the k-th entry in the complete quadrupling system modulo N = 2*n + 1, for n >= 0, with the S(N) = A339049((N-1)/2) cycles of length A053447((N-1)/2) written in row n. See the comment above for QS(N,s(N,i)), i = 1, 2, ..., S(N).

A050976 Haupt-exponents of 4 modulo integers relatively prime to 4.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 2, 4, 9, 3, 11, 10, 9, 14, 5, 5, 6, 18, 6, 10, 7, 6, 23, 21, 4, 26, 10, 9, 29, 30, 3, 6, 33, 11, 35, 9, 10, 15, 39, 27, 41, 4, 14, 11, 6, 5, 18, 24, 15, 50, 51, 6, 53, 18, 18, 14, 22, 6, 12, 55, 10, 50, 7, 7, 65, 9, 18, 34, 69, 23, 30, 14, 21, 74, 15, 12, 10, 26

Offset: 1

Eric W. Weisstein

nonn

Eric Weisstein's World of Mathematics, Multiplicative Order.

Cf. A002326, A002329. Presumably this is a duplicate of A053447.

cp's OEIS Frontend

A070678 Smallest m in range 1..phi(n) such that 7^m == 1 mod n, or 0 if no such number exists.

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Magma

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A070679 Smallest m in range 1..phi(n) such that 9^m == 1 mod n, or 0 if no such number exists.

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Magma

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A070676 Smallest m in range 1..phi(n) such that 3^m == 1 mod n, or 0 if no such number exists.

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Magma

Mathematica

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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Magma

Mathematica

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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A003573 Order of 4 mod 4n+1.

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A003574 Order of 4 mod 4n-1.

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GAP

Maple

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Formula

A302141 Multiplicative order of 16 mod 2n+1.

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Maple

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