%I #35 May 26 2019 19:08:10

%S 2,4,5,8,10,11,17,23,26,28,35,47,53,71,80,82,95,107,143,161,191,215,

%T 242,244,287,323,383,431,485,575,647,728,730,767,863,971,1151,1295,

%U 1457,1535,1727,1943,2186,2188,2303,2591,2915,3071,3455,3887

%N Bases in which 3 is a unique-period prime.

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

%C 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.

%C b is a term if and only if: (a) b = 3^t + 1, t >= 1; (b) b = 2^s*3^t - 1, s >= 0, t >= 1.

%C 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 = 3, there are no nontrivial bases, since ord(3,b) <= 2.

%H Jianing Song, <a href="/A306073/b306073.txt">Table of n, a(n) for n = 1..805</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unique_prime">Unique prime</a>

%e If b = 3^t + 1, t >= 1, then b - 1 only has prime factor 3, so 3 is a unique-period prime in base b.

%e If b = 2^s*3^t - 1, t >= 1, then the prime factors of b^2 - 1 are 3 and prime factors of b - 1 = 2^s*3^t - 2, 3 is the only new prime factor so 3 is a unique-period prime in base b.

%o (PARI)

%o p = 3;

%o gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);

%o test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;

%o 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, ", "))));

%Y Cf. A040017 (unique primes in base 10), A144755 (unique primes in base 2).

%Y Bases in which p is a unique prime: A000051 (p=2), this sequence (p=3), A306074 (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).

%K easy,nonn

%O 1,1

%A _Jianing Song_, Jun 19 2018