A226295 Multiplicative order of the n-th prime modulo the (n+1)th prime.

2, 4, 6, 10, 12, 4, 9, 22, 7, 10, 4, 5, 7, 46, 13, 29, 60, 66, 70, 18, 39, 82, 88, 16, 25, 102, 106, 36, 7, 63, 130, 136, 69, 148, 30, 156, 54, 166, 86, 89, 180, 190, 96, 49, 198, 7, 111, 226, 76, 58, 34, 24, 25, 256, 262, 67, 270, 276, 70, 47, 73, 153, 310

Offset: 1

Author

Emmanuel Vantieghem, Jun 02 2013

Keywords

• nonn

Comments

a(n) is the smallest positive integer m such that p(n)^m == 1 (mod p(n+1)), where p(n) stands for the n-th prime.

Examples

			a(n) = 2 because 2^2 == 1 (mod 3) but 2^1 !== 1 (mod 3); a(6) = 4 because 13^4 == 1 (mod 17) but 13^u !== 1 (mod 17) for u = 1, 2, 3.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. See page 44.
• K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. [Cached copy] See page 44.

Programs

Mathematica

Table[MultiplicativeOrder[Prime[n], Prime[n+1]], {n,80}]

PARI

a(n)=my(p=prime(n));znorder(Mod(p,nextprime(p+1))) \\ Charles R Greathouse IV, Jun 04 2013

Python

from sympy import prime, n_order
def A226295(n): return n_order(prime(n),prime(n+1)) # Chai Wah Wu, Jun 15 2022