A175530 -id:A175530 - cp's OEIS frontend

A299799 Carmichael numbers (A002997) that are Chebyshev pseudoprimes (A175530).

443372888629441, 582920080863121, 39671149333495681, 842526563598720001, 2380296518909971201, 3188618003602886401, 33711266676317630401, 54764632857801026161, 122762671289519184001, 361266866679292635601, 4208895375600667752001, 7673096805497432749441

Offset: 1

Max Alekseyev, Feb 19 2018

Odd composite integer n is in this sequence if n == 1 or p (mod (p^2-1)/2) for every prime p|n.

No other terms below 10^22.

Intersection of A002997 and A175530.

Contains A175531 as a subsequence.

a(9) from Daniel Suteu confirmed and a(10) added by Max Alekseyev, Dec 16 2020

a(11)-a(12) from Max Alekseyev, Apr 21 2024

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.

Original entry on oeis.org

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, #] &]

A175531 Carmichael numbers of order 2.

Original entry on oeis.org

443372888629441, 39671149333495681, 842526563598720001, 2380296518909971201, 3188618003602886401, 4208895375600667752001

Offset: 1

Max Alekseyev, Jun 08 2010

Odd composite integer k is in this sequence if k == 1 or p (mod p^2 - 1) for every prime p|k.

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Everett W. Howe, Higher-order Carmichael numbers, Math. Comp. 69 (2000), 1711-1719. doi:10.1090/S0025-5718-00-01225-4

Subsequence of A002997, A175530, and A299799.

a(6) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

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.

Original entry on oeis.org

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

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).

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

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, A330206, A330208.

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

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

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.

			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, #] &]

A287119 Squarefree composite numbers n such that p^2 - 1 divides n^2 - 1 for every prime p dividing n.

Original entry on oeis.org

8569, 39689, 321265, 430199, 564719, 585311, 608399, 7056721, 11255201, 17966519, 18996769, 74775791, 75669551, 136209151, 321239359, 446660929, 547674049, 866223359, 1068433631, 1227804929, 1291695119, 2315403649, 2585930689, 7229159729, 7809974369, 8117634239

Offset: 1

Thomas Ordowski, May 20 2017

nonn

Such numbers are odd and have at least three prime factors.

Problem: are there infinitely many such numbers?

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002997, A006972, A175530, A175531.

Subsequence of A120944.

isok(n) = {if (issquarefree(n) && !isprime(n), my(f = factor(n)); for (k=1, #f~, if ((n^2-1) % (f[k,1]^2-1), return (0));); return (1););} \\ Michel Marcus, May 20 2017

More terms from Michel Marcus, May 20 2017

a(14)-a(26) from Giovanni Resta, May 20 2017

cp's OEIS Frontend

A299799 Carmichael numbers (A002997) that are Chebyshev pseudoprimes (A175530).

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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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A175531 Carmichael numbers of order 2.

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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.

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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.

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A287119 Squarefree composite numbers n such that p^2 - 1 divides n^2 - 1 for every prime p dividing n.

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