A336663 2-Carmichael numbers: composite numbers n such that A^{n*(n-1)*(n+1)} = I for every matrix A from the group GL(2,Z/nZ).

4, 8, 9, 15, 16, 24, 25, 27, 32, 40, 45, 48, 49, 55, 63, 64, 72, 75, 80, 81, 96, 99, 104, 105, 112, 120, 121, 125, 128, 135, 144, 160, 165, 169, 171, 175, 176, 192, 195, 200, 216, 224, 225, 231, 240, 243, 256, 264, 273, 275, 288, 289, 320, 336, 343, 351, 360

Offset

1,1

Comments

Theorem (an analogue of Korselt's criterion).

For a composite number n the following statements are equivalent:

(1) n is a 2-Carmichael number,

(2) for any prime divisor p of n, (p-1)*(p+1) | n*(n-1)*(n+1).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eugene Karolinsky and Dmytro Seliutin, Carmichael numbers for GL(m), arXiv:2001.10315 [math.NT], 2020.

Mathematica

twoCarmQ[n_] := CompositeQ[n] && AllTrue[FactorInteger[n][[;; , 1]], Divisible[(n - 1)*n*(n + 1), #^2 - 1] &]; Select[Range[360], twoCarmQ] (* Amiram Eldar, Dec 29 2020 *)

Prog

(PARI) is(m) = {my(f=factor(m)[, 1], t=m*(m^2-1)); !isprime(m+(m<2)) && !sum(i=1, #f, t%(f[i]^2-1)); } \\ Jinyuan Wang, Jul 29 2020

Crossrefs

Cf. A002997.

Sequence in context: A137055 A078177 A326692 * A329936 A023886 A158337

Adjacent sequences: A336660 A336661 A336662 * A336664 A336665 A336666

Keyword

nonn

Author

Dmytro Seliutin, Jul 29 2020

Extensions

More terms from Jinyuan Wang, Jul 29 2020

Status

approved