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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. 3
 5719, 6061, 11395, 15841, 17119, 18721, 31535, 67199, 73555, 84419, 117215, 133399, 133951, 174021, 181259, 194833, 226801, 273239, 362881, 469201, 516559, 522899, 534061, 588455, 665281, 700321, 721801, 778261, 903959, 1162349, 1561439, 1708901, 1755001, 1809697 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Amiram Eldar, Table of n, a(n) for n = 1..73 (terms below 10^7) Thøger Bang, Congruence properties of Tchebycheff polynomials, Mathematica Scandinavica, Vol. 2, No. 2 (1955), pp. 327-333, alternative link, Mohamed O. Rayes, Vilmar Trevisan, and Paul S. Wangy, Chebyshev Polynomials and Primality Tests, ICM Technical Report, Kent State University, Kent, Ohio, 1999. See page 8. EXAMPLE 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. MATHEMATICA Select[Range[2*10^4], CompositeQ[#] && Divisible[ChebyshevT[#, 2] - 2, #] && Divisible[ChebyshevT[#, 3] - 3, #] &] CROSSREFS Intersection of A330206 and A330207. Cf. A052155, A053120, A175530. Sequence in context: A244163 A253423 A202376 * A252422 A183647 A028547 Adjacent sequences: A330205 A330206 A330207 * A330209 A330210 A330211 KEYWORD nonn AUTHOR Amiram Eldar, Dec 05 2019 STATUS approved

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Last modified August 11 04:26 EDT 2024. Contains 375059 sequences. (Running on oeis4.)