A226295 -id:A226295 - cp's OEIS frontend

A226367 Multiplicative order of the (n+1)-st prime modulo the n-th prime.

1, 2, 4, 3, 10, 6, 8, 9, 11, 28, 6, 18, 20, 7, 23, 26, 58, 60, 33, 35, 36, 39, 82, 11, 24, 100, 51, 106, 18, 28, 7, 130, 68, 46, 148, 150, 156, 81, 83, 43, 178, 180, 95, 48, 196, 66, 14, 37, 226, 38, 232, 119, 30, 250, 256, 131, 268, 270, 46, 70, 141, 146, 51, 155, 78, 316, 165, 336, 346, 174, 32, 179, 366, 372, 189

Offset: 1

Emmanuel Vantieghem, Jun 05 2013

nonn

a(n) is the smallest positive integer m with the property that p(n+1)^m == 1 (mod p(n)), where p(n) stands for the n-th prime; it is always a divisor of p(n)-1. For n < 10^8, a(n) is never equal to A226295(n).

			a(2) = 2 because 5^2 == 1 (mod 3) but 5^1 !== 1(mod 3).
a(6) = 6  because 17^6 == 1 (mod 13) but 17^u !== 1 (mod 13) for  u < 6.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A226295 (multiplicative order of p(n) mod p(n+1)).

Table[MultiplicativeOrder[Prime[n+1],Prime[n]],{n,1,75}]

vector(80, n, p = prime(n); znorder(Mod(nextprime(p+1), p))) \\ Michel Marcus, Feb 09 2015

A163619 Let q(p) be the smallest prime greater than the prime p. A positive integer n is included in this sequence if n+1 is divisible by q(p) for each prime p dividing n.

Original entry on oeis.org

2, 8, 9, 20, 32, 98, 125, 128, 169, 464, 512, 729, 961, 1280, 2048, 2108, 5252, 8000, 8192, 9728, 15872, 16807, 18176, 22385, 32768, 36992, 50000, 53792, 59049, 78821, 81920, 97556, 98125, 100352, 124659, 131072, 195129, 219488, 223040, 307328

Offset: 1

Leroy Quet, Aug 01 2009

nonn

All terms of this sequence are in sequence A073606.
From Robert Israel, Dec 01 2024: (Start)
If k is a term, then so is k^j for all odd j.
If A226295(k) is even, then prime(k)^(A226295(k)/2) is a term. (End)

			20 is divisible by the primes 2 and 5. q(2) = 3, and q(5)=7. 20+1 = 21 is divisible by both 3 and 7, so 20 is in this sequence.

Cf. A073606, A226295.

filter:= n ->
  andmap(p -> n+1 mod nextprime(p) = 0, numtheory:-factorset(n)):
select(filter, [$2..4*10^5]); # Robert Israel, Dec 01 2024

depQ[n_]:=With[{c=NextPrime[FactorInteger[n][[;;,1]]]},AllTrue[(n+1)/c,IntegerQ]]; Select[Range[ 2,350000],depQ] (* Harvey P. Dale, Jun 10 2023 *)

More terms from Sean A. Irvine, Oct 04 2009

A308510 Prime(k) such that the multiplicative order of prime(k) (mod prime(k+1)) = prime(k+1)-1.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 43, 59, 61, 67, 79, 83, 101, 103, 127, 131, 139, 151, 163, 179, 181, 197, 223, 251, 257, 269, 271, 307, 317, 337, 347, 353, 367, 379, 419, 421, 439, 443, 461, 463, 467, 487, 499, 523, 541, 563, 577, 587, 593, 607, 643, 659, 691, 709, 727, 733, 739

Offset: 1

David James Sycamore, Jun 02 2019

nonn

Prime(k) is a term iff it is a primitive root of prime(k+1). These primes correspond to the records of A226295; if A226295(k) is such a record then prime(k) is a term in this sequence.

			A226295(14) = 46 is a record, so prime(14)=43 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A226295.

Select[Range[740], PrimeQ[#] && MultiplicativeOrder[#, p=NextPrime[#]] == p-1 &] (* Amiram Eldar, Jul 04 2019 *)

isok(p) = isprime(p) && (q=nextprime(p+1)) && (znorder(Mod(p, q)) == q-1) \\ Michel Marcus, Jun 02 2019

More terms from Michel Marcus, Jun 02 2019

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)).

A381924 Multiplicative order of n mod prime(n).

Original entry on oeis.org

1, 2, 4, 3, 5, 12, 16, 6, 11, 28, 30, 9, 40, 21, 46, 13, 29, 60, 33, 7, 24, 13, 41, 88, 48, 100, 34, 106, 54, 7, 63, 26, 136, 23, 74, 75, 39, 9, 166, 86, 178, 5, 95, 192, 196, 99, 105, 222, 113, 228, 29, 34, 120, 250, 256, 262, 67, 270, 46, 8, 47, 292, 153, 155, 312

Offset: 1

Giorgos Kalogeropoulos, Mar 12 2025

nonn

a(n) is the least k such that prime(n) divides n^k-1.

			a(12) = 9 because the multiplicative order of 12 mod prime(12) is 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..3500

Cf. A226295, A014664, A091185 (n^(-1) mod prime(n)).

[Order(n, NthPrime(n)) : n in [1..65]]; // Vincenzo Librandi, Mar 25 2025

Table[MultiplicativeOrder[n,Prime[n]],{n,65}]

a(n) = znorder(Mod(n, prime(n))); \\ Michel Marcus, Mar 12 2025

If n is a primitive root modulo prime(n), a(n) = prime(n) - 1.

cp's OEIS Frontend

A226367 Multiplicative order of the (n+1)-st prime modulo the n-th prime.

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A163619 Let q(p) be the smallest prime greater than the prime p. A positive integer n is included in this sequence if n+1 is divisible by q(p) for each prime p dividing n.

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A308510 Prime(k) such that the multiplicative order of prime(k) (mod prime(k+1)) = prime(k+1)-1.

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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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A381924 Multiplicative order of n mod prime(n).

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