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A330207 Chebyshev pseudoprimes to base 3: composite numbers k such that T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.

%I A330207 #18 Feb 16 2025 08:33:59
%S A330207 14,35,119,169,385,434,574,741,779,899,935,961,1105,1106,1121,1189,
%T A330207 1443,1479,2001,2419,2555,2915,3059,3107,3383,3605,3689,3741,3781,
%U A330207 3827,4199,4795,4879,4901,5719,6061,6083,6215,6265,6441,6479,6601,6895,6929,6931,6965
%N A330207 Chebyshev pseudoprimes to base 3: composite numbers k such that T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.
%C A330207 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).
%H A330207 Amiram Eldar, <a href="/A330207/b330207.txt">Table of n, a(n) for n = 1..1000</a>
%H A330207 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 A330207 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 A330207 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 A330207 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev Polynomial of the First Kind</a>.
%H A330207 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%e A330207 14 is in the sequence since it is composite and T(14, 3) = 26102926097 == 3 (mod 14).
%t A330207 Select[Range[1000], CompositeQ[#] && Divisible[ChebyshevT[#, 3] - 3, #] &]
%Y A330207 Cf. A053120, A175530, A330206, A330208.
%K A330207 nonn
%O A330207 1,1
%A A330207 _Amiram Eldar_, Dec 05 2019