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.

14, 35, 119, 169, 385, 434, 574, 741, 779, 899, 935, 961, 1105, 1106, 1121, 1189, 1443, 1479, 2001, 2419, 2555, 2915, 3059, 3107, 3383, 3605, 3689, 3741, 3781, 3827, 4199, 4795, 4879, 4901, 5719, 6061, 6083, 6215, 6265, 6441, 6479, 6601, 6895, 6929, 6931, 6965

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).

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

14 is in the sequence since it is composite and T(14, 3) = 26102926097 == 3 (mod 14).

Mathematica

Select[Range[1000], CompositeQ[#] && Divisible[ChebyshevT[#, 3] - 3, #] &]

Crossrefs

Cf. A053120, A175530, A330206, A330208.

Sequence in context: A159242 A054309 A321135 * A104317 A178933 A203803

Adjacent sequences: A330204 A330205 A330206 * A330208 A330209 A330210

Keyword

nonn

Author

Amiram Eldar, Dec 05 2019

Status

approved