OFFSET
1,2
COMMENTS
Let {Zs(m, 10, 1)} be the Zsigmondy numbers for a = 10, b = 1: Zs(m, 10, 1) is the greatest divisor of 10^m - 1^m that is coprime to 10^r - 1^r for all positive integers r < m. Then this sequence gives m such that Zs(m, 10, 1) is a prime power (e.g., Zs(1, 10, 1) = 9 = 3^2, Zs(2, 10, 1) = 11, Zs(3, 10, 1) = 37, Zs(4, 10, 1) = 101). It is very likely that Zs(m, 10, 1) is prime if m > 1 is in this sequence (note that the Mathematica and PARI programs below are based on this assumption). - Jianing Song, Aug 12 2020
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..106
Chris K. Caldwell, Unique (period) primes and the factorization of cyclotomic polynomials minus one, Mathematica Japonica, 26 (1997), 189-195.
Chris K. Caldwell & H. Dubner, Unique-Period Primes, Table 2 in Journal of Recreational Mathematics 29(1) 46 1998.
Robert G. Wilson v, Notes, n.d.
MATHEMATICA
lst={1}; Do[p=Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], AppendTo[lst, n]], {n, 3000}]; lst (* T. D. Noe, Sep 08 2005 *)
PROG
(PARI) isok(n) = if (n==1, 1, my(p = polcyclo(n, 10)); isprime(p/gcd(p, n))); \\ Michel Marcus, Jun 20 2018
CROSSREFS
KEYWORD
nonn,nice,base
AUTHOR
EXTENSIONS
More terms from T. D. Noe, Sep 08 2005
a(48)-a(52) from Ray Chandler, Jul 09 2008
STATUS
approved