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Chebyshev pseudoprimes to base 2: composite numbers k such that T(k, 2) == 2 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.
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%I #13 Dec 06 2019 10:29:21

%S 209,231,399,455,901,903,923,989,1295,1729,1855,2015,2211,2345,2639,

%T 2701,2795,2911,3007,3201,3439,3535,3801,4823,5291,5719,6061,6767,

%U 6989,7421,8569,9503,9591,9869,10439,10609,11041,11395,11951,11991,13133,13529,13735,13871

%N Chebyshev pseudoprimes to base 2: composite numbers k such that T(k, 2) == 2 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.

%C Bang proved that T(p, a) == a (mod p) for every a > 0 and every odd prime. Rayes et al. (1999) defined Chebyshev pseudoprimes to base a as composite numbers k such that T(k, a) == a (mod k). They noted that the first Chebyshev pseudoprime to base 2 is 209.

%H Amiram Eldar, <a href="/A330206/b330206.txt">Table of n, a(n) for n = 1..1000</a>

%H Thøger Bang, <a href="https://www.jstor.org/stable/24489044">Congruence properties of Tchebycheff polynomials</a>, Mathematica Scandinavica, Vol. 2, No. 2 (1955), pp. 327-333, <a href="https://www.mscand.dk/article/download/10418/8439">alternative link</a>,

%H David Pokrass Jacobs, Mohamed O. Rayes, and Vilmar Trevisan. <a href="http://mi.mathnet.ru/eng/adm159">Characterization of Chebyshev Numbers</a>, Algebra and Discrete Mathematics, Vol. 2 (2008), pp. 65-82.

%H Mohamed O. Rayes, Vilmar Trevisan, and Paul S. Wangy, <a href="http://icm.mcs.kent.edu/reports/1999/chebpol.pdf">Chebyshev Polynomials and Primality Tests</a>, ICM Technical Report, Kent State University, Kent, Ohio, 1999. See page 8.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev Polynomial of the First Kind</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%e 209 is in the sequence since 209 = 11 * 19 is composite and T(209, 2) - 2 is divisible by 209.

%t Select[Range[15000], CompositeQ[#] && Divisible[ChebyshevT[#, 2] - 2, #] &]

%Y Cf. A053120, A175530, A330207, A330208.

%K nonn

%O 1,1

%A _Amiram Eldar_, Dec 05 2019