A175531 -id:A175531 - cp's OEIS frontend

A175530 Pseudoprime Chebyshev numbers: odd composite integers n such that T_n(a) == a (mod n) for all integers a, where T(x) is Chebyshev polynomial of first kind.

7056721, 79397009999, 443372888629441, 582920080863121, 2491924062668039, 14522256850701599, 39671149333495681, 242208715337316001, 729921147126771599, 842526563598720001, 1881405190466524799, 2380296518909971201, 3188618003602886401, 33711266676317630401, 54764632857801026161, 55470688965343048319, 72631455338727028799, 122762671289519184001, 361266866679292635601, 734097107648270852639

Offset: 1

Max Alekseyev, Jun 08 2010

Odd composite integer n is a pseudoprime Chebyshev number iff the n-th term of Lucas sequence satisfies the congruence V_n(P,1) == P (mod n) for any integer P.
Odd composite integer n is a pseudoprime Chebyshev number iff n == +1 or -1 (mod p-1) and n == +1 or -1 (mod p+1) for each prime p|n.
No other terms below 10^21.
Named after the Russian mathematician Pafnuty Chebyshev (1821-1894) after whom the "Chebyshev polynomials" were also named. - Amiram Eldar, Jun 15 2021

			7056721 = 7 * 47 * 89 * 241, while 7056721 == 1 (mod 7-1), == 1 (mod 7+1), == -1 (mod 47-1), == 1 (mod 47+1), == 1 (mod 89-1), == 1 (mod 89+1), == 1 (mod 241-1), and == 1 (mod 241+1).

David Broadhurst, The second Chebyshev number, NMBRTHRY Mailing List, 4 June 2010.
Kok Seng Chua, Chebyshev polynomials and higher order Lucas Lehmer algorithm, arXiv:2010.02677 [math.NT], 2020. Mentions this sequence.
David Pokrass Jacobs, Mohamed O. Rayes, and Vilmar Trevisan, Characterization of Chebyshev Numbers, Algebra and Discrete Mathematics, Vol. 2 (2008), pp. 65-82.
Eric Weisstein's World of Mathematics, Lucas Sequence.

Terms that are Carmichael numbers (A002997) form A299799.
Contains A175531 as a subsequence.
Cf. A053120

a(1) is given in the Jacobs-Rayes-Trevisan paper.
a(2) from Kevin Acres, David Broadhurst, Ray Chandler, David Cleaver, Mike Oakes, and Christ van Willegen, Jun 04 2010
a(3)-a(20) from Max Alekseyev, Jun 08 2010, Feb 26 2018, Dec 16 2020

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

Original entry on oeis.org

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

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

A175530 Pseudoprime Chebyshev numbers: odd composite integers n such that T_n(a) == a (mod n) for all integers a, where T(x) is Chebyshev polynomial of first kind.

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

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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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