A249330 -id:A249330 - cp's OEIS frontend

A072859 Primes p for which the period of 1/p is prime.

11, 37, 41, 53, 79, 83, 107, 173, 227, 239, 271, 317, 347, 359, 467, 479, 563, 587, 643, 719, 733, 773, 797, 839, 907, 1031, 1187, 1231, 1283, 1307, 1319, 1439, 1493, 1523, 1627, 1637, 1879, 1907, 1987, 2027, 2039, 2467, 2477, 2677, 2791, 2837, 2879, 2963

Offset: 1

Benoit Cloitre, Jul 26 2002

Primes p such that the multiplicative order of 10 (mod p) is prime. - Joerg Arndt, Oct 26 2014

Together with 3, complement in primes of A249330. - Arkadiusz Wesolowski, Oct 25 2014

			1/37 = 0. 027 027 ... with period = 3, hence 37 is in the sequence.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A002371, A249330.

Select[Prime[Range[500]],PrimeQ[MultiplicativeOrder[10,#]]&] (* Ray Chandler, Oct 31 2011 *)

f(n)=if(n<4,n==2,znorder(Mod(10, prime(n))));
for(n=1,100,if(isprime(f(n))==1,print1(prime(n),", ")))

A386519 Index of the smallest prime p such that the number of digits L in the repeating decimal period of 1/p equals the n-th prime.

Original entry on oeis.org

5, 12, 13, 52, 2431, 16, 153888, 27417323062119920, 223378173194137397198, 452, 406, 150886, 23, 40, 2153717, 28, 92971458509, 130, 40998

Offset: 1

Jean-Marc Rebert, Jul 24 2025

In general, for (q,2*5)=1, the length of the period of 1/q is equal to the multiplicative order of 10 modulo q, which is the smallest k such that 10^k == 1 (mod q). It follows that a(n) must be a prime divisor of 10^prime(n)-1. Hence, apart from a(2), we have prime(a(n)) = A147555(n) and a(20) is the index of the prime 241573142393627673576957439049. - Giovanni Resta, Jul 24 2025

			a(1) = 5, since the 5th prime, p = 11, has a repeating decimal period of length L = 2, and 2 = prime(1). There is no smaller prime for which the period length equals the 1st prime.
 n      a(n)         p  L
 1         5        11  2
 2        12        37  3
 3        13        41  5
 4        52       239  7
 5      2431     21649 11
 6        16        53 13
 7    153888   2071723 17

Cf. A072859, A002371, A249330, A147555.

a(8)-a(19) from Giovanni Resta, Jul 24 2025

cp's OEIS Frontend

A072859 Primes p for which the period of 1/p is prime.

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