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

1,1

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 the first Chebyshev pseudoprime to base 2 is 209.

Links

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.

Example

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

Mathematica

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

Crossrefs

Cf. A053120, A175530, A330207, A330208.

Sequence in context: A080532 A153442 A159274 * A025334 A025326 A060979

Adjacent sequences: A330203 A330204 A330205 * A330207 A330208 A330209

Keyword

nonn

Author

Amiram Eldar, Dec 05 2019

Status

approved