

A306073


Bases in which 3 is a uniqueperiod prime.


5



2, 4, 5, 8, 10, 11, 17, 23, 26, 28, 35, 47, 53, 71, 80, 82, 95, 107, 143, 161, 191, 215, 242, 244, 287, 323, 383, 431, 485, 575, 647, 728, 730, 767, 863, 971, 1151, 1295, 1457, 1535, 1727, 1943, 2186, 2188, 2303, 2591, 2915, 3071, 3455, 3887
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OFFSET

1,1


COMMENTS

A prime p is called a uniqueperiod prime in base b if there is no other prime q such that the period length of the baseb expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a uniqueperiod 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 = 3^t + 1, t >= 1; (b) b = 2^s*3^t  1, s >= 0, t >= 1.
For every odd prime p, p is a uniqueperiod 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 uniqueperiod 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 uniqueperiod prime, with ord(b,p) >= 3. For p = 3, there are no nontrivial bases, since ord(3,b) <= 2.


LINKS

Jianing Song, Table of n, a(n) for n = 1..805
Wikipedia, Unique prime


EXAMPLE

If b = 3^t + 1, t >= 1, then b  1 only has prime factor 3, so 3 is a uniqueperiod prime in base b.
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 uniqueperiod prime in base b.


PROG

(PARI)
p = 3;
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, ", "))));


CROSSREFS

Cf. A040017 (unique primes in base 10), A144755 (unique primes in base 2).
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).
Sequence in context: A018699 A073628 A067938 * A018457 A046809 A112777
Adjacent sequences: A306070 A306071 A306072 * A306074 A306075 A306076


KEYWORD

easy,nonn


AUTHOR

Jianing Song, Jun 19 2018


STATUS

approved



