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%I M0592 #44 Oct 15 2022 15:22:59
%S 1,2,3,4,9,10,12,14,19,23,24,36,38,39,48,62,93,106,120,134,150,196,
%T 294,317,320,385,586,597,654,738,945,1031,1172,1282,1404,1426,1452,
%U 1521,1752,1812,1836,1844,1862,2134,2232,2264,2667,3750,3903,3927,4274,4354
%N Unique period lengths of primes mentioned in A007615.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.
%H Ray Chandler, <a href="/A007498/b007498.txt">Table of n, a(n) for n = 1..106</a>
%H Chris K. Caldwell, <a href="http://docplayer.net/43219082-Unique-period-primes-and-the-factorization-of-cyclotomic-polynomials-minus-one.html">Unique (period) primes and the factorization of cyclotomic polynomials minus one</a>, Mathematica Japonica, 26 (1997), 189-195.
%H Chris K. Caldwell & H. Dubner, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.693.5325">Unique-Period Primes</a>, Table 2 in Journal of Recreational Mathematics 29(1) 46 1998.
%H Robert G. Wilson v, <a href="/A007498/a007498.pdf">Notes, n.d.</a>
%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%t 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 *)
%o (PARI) isok(n) = if (n==1, 1, my(p = polcyclo(n, 10)); isprime(p/gcd(p, n))); \\ _Michel Marcus_, Jun 20 2018
%Y Cf. A007615, A002371, A048595, A006883, A007732, A051626, A046108.
%Y Cf. A161508 (unique period lengths in base 2).
%K nonn,nice,base
%O 1,2
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E More terms from _T. D. Noe_, Sep 08 2005
%E a(48)-a(52) from _Ray Chandler_, Jul 09 2008
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