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A330208 Chebyshev pseudoprimes to both base 2 and base 3: composite numbers k such that T(k, 2) == 2 (mod k) and T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.

%I A330208 #23 Dec 10 2019 04:04:08
%S A330208 5719,6061,11395,15841,17119,18721,31535,67199,73555,84419,117215,
%T A330208 133399,133951,174021,181259,194833,226801,273239,362881,469201,
%U A330208 516559,522899,534061,588455,665281,700321,721801,778261,903959,1162349,1561439,1708901,1755001,1809697
%N A330208 Chebyshev pseudoprimes to both base 2 and base 3: composite numbers k such that T(k, 2) == 2 (mod k) and T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.
%C A330208 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 there are no Chebyshev pseudoprimes in both bases 2 and 3 below 2000.
%H A330208 Amiram Eldar, <a href="/A330208/b330208.txt">Table of n, a(n) for n = 1..73</a> (terms below 10^7)
%H A330208 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 A330208 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.
%e A330208 5719 is in the sequence since 5719 = 7 * 19 * 43 is composite and both T(5719 , 2) - 2 and T(5719, 3) - 3 are divisible by 5719.
%t A330208 Select[Range[2*10^4], CompositeQ[#] && Divisible[ChebyshevT[#, 2] - 2, #] && Divisible[ChebyshevT[#, 3] - 3, #] &]
%Y A330208 Intersection of A330206 and A330207.
%Y A330208 Cf. A052155, A053120, A175530.
%K A330208 nonn
%O A330208 1,1
%A A330208 _Amiram Eldar_, Dec 05 2019