A330208 - OEIS

%I #23 Dec 10 2019 04:04:08

%S 5719,6061,11395,15841,17119,18721,31535,67199,73555,84419,117215,

%T 133399,133951,174021,181259,194833,226801,273239,362881,469201,

%U 516559,522899,534061,588455,665281,700321,721801,778261,903959,1162349,1561439,1708901,1755001,1809697

%N 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.

%C 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.

%H Amiram Eldar, <a href="/A330208/b330208.txt">Table of n, a(n) for n = 1..73</a> (terms below 10^7)

%H Thøger Bang, <a href="https://www.jstor.org/stable/24489044">Congruence properties of Tchebycheff polynomials</a>, Mathematica Scandinavica, Vol. 2, No. 2 (1955), pp. 327-333, <a href="https://www.mscand.dk/article/download/10418/8439">alternative link</a>,

%H Mohamed O. Rayes, Vilmar Trevisan, and Paul S. Wangy, <a href="http://icm.mcs.kent.edu/reports/1999/chebpol.pdf">Chebyshev Polynomials and Primality Tests</a>, ICM Technical Report, Kent State University, Kent, Ohio, 1999. See page 8.

%e 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.

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

%Y Intersection of A330206 and A330207.

%Y Cf. A052155, A053120, A175530.

%K nonn

%O 1,1

%A _Amiram Eldar_, Dec 05 2019