A330207 -id:A330207 - cp's OEIS frontend

A330206 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.

209, 231, 399, 455, 901, 903, 923, 989, 1295, 1729, 1855, 2015, 2211, 2345, 2639, 2701, 2795, 2911, 3007, 3201, 3439, 3535, 3801, 4823, 5291, 5719, 6061, 6767, 6989, 7421, 8569, 9503, 9591, 9869, 10439, 10609, 11041, 11395, 11951, 11991, 13133, 13529, 13735, 13871

Offset: 1

Amiram Eldar, Dec 05 2019

nonn

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..1000
Thøger Bang, Congruence properties of Tchebycheff polynomials, Mathematica Scandinavica, Vol. 2, No. 2 (1955), pp. 327-333, alternative link,
David Pokrass Jacobs, Mohamed O. Rayes, and Vilmar Trevisan. Characterization of Chebyshev Numbers, Algebra and Discrete Mathematics, Vol. 2 (2008), pp. 65-82.
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.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
Wikipedia, Chebyshev polynomials.

Cf. A053120, A175530, A330207, A330208.

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

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.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 05 2019

nonn

			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.

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.

Intersection of A330206 and A330207.

Cf. A052155, A053120, A175530.

Select[Range[2*10^4], CompositeQ[#] && Divisible[ChebyshevT[#, 2] - 2, #] && Divisible[ChebyshevT[#, 3] - 3, #] &]

cp's OEIS Frontend

A330206 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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