A306077 Bases in which 13 is a unique-period prime.

2, 3, 4, 5, 12, 14, 22, 23, 25, 51, 103, 168, 170, 207, 239, 337, 415, 675, 831, 1351, 1663, 2196, 2198, 2703, 3327, 4393, 5407, 6655, 8787, 10815, 13311, 17575, 21631, 26623, 28560, 28562, 35151, 43263, 53247, 57121, 70303, 86527, 106495, 114243, 140607, 173055, 212991, 228487, 281215, 346111

Offset: 1

Author

Jianing Song, Jun 19 2018

Keywords

• easy
• nonn

Comments

A prime p is called a unique-period prime in base b if there is no other prime q such that the period of the base-b expansion of its reciprocal, 1/p, is equal to the period of the reciprocal of q, 1/q.

A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.

b is a term if and only if: (a) b = 13^t + 1, t >= 1; (b) b = 2^s*13^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 4, 5, 22, 23, 239.

For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 13, the nontrivial bases are 2, 3, 4, 5, 22, 23, 239.

Examples

			1/13 has period 12 in base 2. Note that 3, 5, 7, 13, 31 are the only prime factors of 2^12 - 1 = 4095, but 1/3 has period 2, 1/5 has period 4, 1/7 has period 3, 1/31 has period 5, so 13 is a unique-period prime in base 2. (For the same reason, 13 is a unique-period prime in base 4.)
1/13 has period 3 in base 3. Note that 2, 13 are the only prime factors of 3^3 - 1 = 26, but 1/2 has period 1, so 13 is a unique-period prime in base 3.
1/13 has period 3 in base 22. Note that 3, 7, 13 are the only prime factors of 22^3 - 1 = 10647, but 1/3 and 1/7 both have period 1, so 13 is a unique-period prime in base 22.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..554
• Wikipedia, Unique prime

Crossrefs

Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), A306074 (p=5), A306075 (p=7), A306076 (p=11), this sequence (p=13).

Programs

PARI

p = 13;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));