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%I #27 Jun 28 2018 05:57:31
%S 2,3,4,6,7,9,19,24,26,39,49,79,99,124,126,159,199,249,319,399,499,624,
%T 626,639,799,999,1249,1279,1599,1999,2499,2559,3124,3126,3199,3999,
%U 4999,5119,6249,6399,7999,9999,10239,12499,12799,15624,15626,15999,19999,20479
%N Bases in which 5 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 = 5^t + 1, t >= 1; (b) b = 2^s*5^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 7.
%C For every odd prime p, p is a 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 = 5, the nontrivial bases are 2, 3, 7.
%H Jianing Song, <a href="/A306074/b306074.txt">Table of n, a(n) for n = 1..544</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unique_prime">Unique prime</a>
%e 1/5 has period length 4 in base 2. Note that 3 and 5 are the only prime factors of 2^4 - 1 = 15, but 1/3 has period length 2, so 5 is a unique-period prime in base 2.
%e 1/5 has period length 4 in base 3. Note that 2 and 5 are the only prime factors of 3^4 - 1 = 80, but 1/2 has period length 1, so 5 is a unique-period prime in base 3.
%e 1/5 has period length 4 in base 7. Note that 2, 3 and 5 are the only prime factors of 7^4 - 1 = 2400, but 1/2 and 1/3 both have period length 1, so 5 is a unique-period prime in base 7.
%o (PARI)
%o p = 5;
%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-period primes in base 10), A144755 (base 2).
%Y Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), this sequence (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).
%K easy,nonn
%O 1,1
%A _Jianing Song_, Jun 19 2018
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