

A007498


Unique period lengths of primes mentioned in A007615.
(Formerly M0592)


11



1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, 4274, 4354
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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), 2224.


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), 189195.
Chris K. Caldwell & H. Dubner, UniquePeriod Primes, Table 2 in Journal of Recreational Mathematics 29(1) 46 1998.
Robert G. Wilson v, Notes, n.d.
Index entries for sequences related to decimal expansion of 1/n


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

Cf. A007615, A002371, A048595, A006883, A007732, A051626.
Cf. A161508 (unique period lengths in base 2).
Sequence in context: A294485 A332772 A084368 * A073338 A200260 A190119
Adjacent sequences: A007495 A007496 A007497 * A007499 A007500 A007501


KEYWORD

nonn,nice,base


AUTHOR

N. J. A. Sloane, Robert G. Wilson v


EXTENSIONS

More terms from T. D. Noe, Sep 08 2005
a(48)a(52) from Ray Chandler, Jul 09 2008


STATUS

approved



