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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. 3
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 (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).
LINKS
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.
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
Sequence in context: A159242 A054309 A321135 * A104317 A178933 A203803
KEYWORD
nonn
AUTHOR
Amiram Eldar, Dec 05 2019
STATUS
approved

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Last modified April 17 13:58 EDT 2024. Contains 371764 sequences. (Running on oeis4.)