A211241 -id:A211241 - cp's OEIS frontend

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

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.

A305331 Multiplicative order of 5 (mod p^2), where p = prime(n), or 0 if 5 and p are not coprime.

Original entry on oeis.org

1, 6, 0, 42, 55, 52, 272, 171, 506, 406, 93, 1332, 820, 1806, 2162, 2756, 1711, 1830, 1474, 355, 5256, 3081, 6806, 3916, 9312, 2525, 10506, 11342, 2943, 12656, 5334, 8515, 18632, 9591, 5513, 11325, 24492, 8802, 27722, 29756, 15931, 2715, 3629, 37056, 38612

Offset: 1

Felix Fröhlich, May 30 2018

nonn

Cf. A211241, A305332, A305333.

Table[If[p==5, 0, MultiplicativeOrder[5, p^2]], {p, Prime@ Range@ 45}] (* Giovanni Resta, May 31 2018 *)

a(n) = my(p=prime(n)); if(p==5, return(0), return(znorder(Mod(5, p^2))))

A305332 Multiplicative order of 5 (mod A123692(n)^2).

Original entry on oeis.org

1, 10385, 40486, 13367790, 1645333506, 6692367336, 11796759175

Offset: 1

Felix Fröhlich, May 30 2018

From Eric Chen, Jun 07 2018: (Start)
    b    known Wieferich primes in base b (multiplicative order of b mod these primes (also these primes^2)) (if the order is p-1, then b is a primitive root to mod this prime (but not mod this prime^2), see A055578)
  1093 (364), 3511 (1755)
  11 (5), 1006003 (1006002)
  1093 (182), 3511 (1755)
  2 (1), 20771 (10385), 40487 (40486), 53471161 (13367790), 1645333507 (1645333506), 6692367337 (6692367336), 188748146801 (11796759175)
  66161 (66160), 534851 (106970), 3152573 (788143)
  5 (4), 491531 (245765)
  3 (2), 1093 (364), 3511 (585)
  2 (1), 11 (5), 1006003 (503001)
  3 (1), 487 (486), 56598313 (56598312)
  71 (70)
  2693 (2692), 123653 (123652)
  2 (1), 863 (862), 1747591 (873795)
  29 (28), 353 (352), 7596952219 (7596952218)
  29131 (29130), 119327070011 (59663535005)
  1093 (91), 3511 (1755)
  2 (1), 3 (2), 46021 (7670), 48947 (24473), 478225523351 (478225523350)
  5 (4), 7 (3), 37 (36), 331 (110), 33923 (33922), 1284043 (428014)
  3 (1), 7 (6), 13 (12), 43 (42), 137 (68), 63061489 (63061488)
  281 (140), 46457 (46456), 9377747 (9377746), 122959073 (122959072)
  2 (1)
  13 (3), 673 (224), 1595813 (797906), 492366587 (246183293), 9809862296159 (44999368331)
  13 (6), 2481757 (827252), 13703077 (13703076), 15546404183 (7773202091), 2549536629329 (2549536629328)
  5 (2), 25633 (6408)
These orders n will satisfy that Phi_n(b) is divisible by p^2, where Phi is the cyclotomic polynomial. (Usually, Phi_n(b) is squarefree, but these are all exceptions; i.e., if p^2 divides Phi_n(b) (except the case p = 2, n = 2 and b == 3 (mod 4)), then p is a Wieferich prime in base b.)
(End)

Cf. A123692, A211241, A305331, A305333.

v=[2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801]; for(k=1, #v, print1(znorder(Mod(5, v[k]^2)), ", "))

a(n) = A305331(A123692(n)).

A305333 Let p be the n-th base-5 Wieferich prime (A123692). a(n) is the value of p-1 divided by the multiplicative order of 5 (mod p^2).

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 16

Offset: 1

Felix Fröhlich, May 30 2018

Cf. A123692, A211241, A305331, A305332.

v=[2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801]; for(k=1, #v, my(ord=znorder(Mod(5, v[k]^2))); print1((v[k]-1)/ord, ", "))

a(n) = (A123692(n)-1)/A305332(n).

A383411 Primes p such that gcd(ord_p(3), ord_p(5)) = 1.

Original entry on oeis.org

2, 13, 313, 51169, 797161, 3482851, 5096867, 12207031, 162410641, 368385827, 1001523179, 4902814883, 104849105869, 131772143257, 572027881891

Offset: 1

Li GAN, Apr 26 2025

'ord_p' here means the multiplicative order modulo p, not to be confused with the p-adic order that is also often denoted by ord_p.

Mathematics Stack Exchange, For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?, 2021.

Cf. A062117, A211241, A344202.

Select[Range[10000], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[3, #], MultiplicativeOrder[5, #]] &]

forprime(p=13,oo,if(1==gcd(znorder(Mod(5,p)),znorder(Mod(3,p))),print1(p,", "))); \\ Joerg Arndt, Apr 26 2025

a(13)-a(15) from Bill McEachen, May 11 2025

A240660 Least k such that 5^k == -1 (mod prime(n)), or 0 if no such k exists.

Original entry on oeis.org

1, 1, 0, 3, 0, 2, 8, 0, 11, 7, 0, 18, 10, 21, 23, 26, 0, 15, 11, 0, 36, 0, 41, 22, 48, 0, 51, 53, 0, 56, 21, 0, 68, 0, 0, 0, 78, 27, 83, 86, 0, 0, 0, 96, 98, 0, 0, 111, 113, 57, 116, 0, 20, 0, 128, 131, 0, 0, 138, 70, 141, 146, 153, 0, 4, 158, 0, 56, 173, 87

Offset: 1

T. D. Noe, Apr 14 2014

nonn

The least k, if it exists, such that prime(n) divides 5^k + 1.

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

Cf. A211241 (order of 5 mod prime(n)).

Table[p = Prime[n]; s = Select[Range[p/2], PowerMod[5, #, p] == p - 1 &, 1]; If[s == {}, 0, s[[1]]], {n, 100}]

a(1) = 1; for n > 1, a(n) = A211241(n)/2 if A211241(n) is even, otherwise 0.

A323376 Square array read by ascending antidiagonals: T(n,k) is the multiplicative order of the n-th prime modulo the k-th prime, or 0 if n = k, n >= 1, k >= 1.

Original entry on oeis.org

0, 1, 2, 1, 0, 4, 1, 2, 4, 3, 1, 1, 0, 6, 10, 1, 2, 4, 6, 5, 12, 1, 1, 1, 0, 5, 3, 8, 1, 2, 4, 3, 10, 4, 16, 18, 1, 1, 4, 2, 0, 12, 16, 18, 11, 1, 2, 2, 6, 10, 12, 16, 9, 11, 28, 1, 2, 4, 6, 10, 0, 16, 3, 22, 28, 5, 1, 1, 2, 3, 10, 6, 4, 3, 22, 14, 30, 36

Offset: 1

Jianing Song, Jan 12 2019

The maximum element in the k-th column is prime(k) - 1. By Dirichlet's theorem on arithmetic progressions, all divisors of prime(k) - 1 occur infinitely many times in the n-th column.

			Table begins
     |  k  | 1  2  3  4   5   6   7   8   9  10  ...
   n | p() | 2  3  5  7  11  13  17  19  23  29  ...
  ---+-----+----------------------------------------
   1 |   2 | 0, 2, 4, 3, 10, 12,  8, 18, 11, 28, ...
   2 |   3 | 1, 0, 4, 6,  5,  3, 16, 18, 11, 28, ...
   3 |   5 | 1, 2, 0, 6,  5,  4, 16,  9, 22, 14, ...
   4 |   7 | 1, 1, 4, 0, 10, 12, 16,  3, 22,  7, ...
   5 |  11 | 1, 2, 1, 3,  0, 12, 16,  3, 22, 28, ...
   6 |  13 | 1, 1, 4, 2, 10,  0,  4, 18, 11, 14, ...
   7 |  17 | 1, 2, 4, 6, 10,  6,  0,  9, 22,  4, ...
   8 |  19 | 1, 1, 2, 6, 10, 12,  8,  0, 22, 28, ...
   9 |  23 | 1, 2, 4, 3,  1,  6, 16,  9 , 0,  7, ...
  10 |  29 | 1, 2, 2, 1, 10,  3, 16, 18, 11,  0, ...
  ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened

Cf. A250211.
Cf. A014664 (1st row), A062117 (2nd row), A211241 (3rd row), A211243 (4th row), A039701 (2nd column).
Cf. A226367 (lower diagonal), A226295 (upper diagonal).

A:= (n, k)-> `if`(n=k, 0, (p-> numtheory[order](p(n), p(k)))(ithprime)):
seq(seq(A(1+d-k, k), k=1..d), d=1..14);  # Alois P. Heinz, Feb 06 2019

T[n_, k_] := If[n == k, 0, MultiplicativeOrder[Prime[n], Prime[k]]];Table[T[n, k], {n, 1, 10}, {k, 1, 10}] (* Peter Luschny, Jan 20 2019 *)

T(n,k) = if(n==k, 0, znorder(Mod(prime(n), prime(k))))

T(n,k) = A250211(prime(n), prime(k)).

A382414 Primes p such that gcd(ord_p(2), ord_p(5)) = 1.

Original entry on oeis.org

31, 601, 2593, 599479, 204700049, 466344409, 668731841, 11638603429

Offset: 1

Li GAN, Apr 26 2025

'ord_p' here means the multiplicative order modulo p, not to be confused with the p-adic order that is also often denoted by ord_p.

1790799748670521, 58523123221688392679 and 14551915228363037109375001 are also terms. - Giorgos Kalogeropoulos, May 03 2025

Mathematics Stack Exchange, For all alpha,beta in N, are there only finitely many primes so that gcd(ordp(alpha),ordp(beta))=1?, 2021

Cf. A014664, A211241, A344202, A383411.

Select[Range[10000], PrimeQ[#] && CoprimeQ[MultiplicativeOrder[2, #], MultiplicativeOrder[5, #]] &]

forprime(p=13, oo, if(1==gcd(znorder(Mod(5, p)), znorder(Mod(2, p))), print1(p, ", "))); \\ Joerg Arndt, Apr 26 2025

cp's OEIS Frontend

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

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A305331 Multiplicative order of 5 (mod p^2), where p = prime(n), or 0 if 5 and p are not coprime.

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A305332 Multiplicative order of 5 (mod A123692(n)^2).

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A305333 Let p be the n-th base-5 Wieferich prime (A123692). a(n) is the value of p-1 divided by the multiplicative order of 5 (mod p^2).

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A383411 Primes p such that gcd(ord_p(3), ord_p(5)) = 1.

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A240660 Least k such that 5^k == -1 (mod prime(n)), or 0 if no such k exists.

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A323376 Square array read by ascending antidiagonals: T(n,k) is the multiplicative order of the n-th prime modulo the k-th prime, or 0 if n = k, n >= 1, k >= 1.

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A382414 Primes p such that gcd(ord_p(2), ord_p(5)) = 1.

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