Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 Dec 29 2020 02:53:07
%S 4,8,9,15,16,24,25,27,32,40,45,48,49,55,63,64,72,75,80,81,96,99,104,
%T 105,112,120,121,125,128,135,144,160,165,169,171,175,176,192,195,200,
%U 216,224,225,231,240,243,256,264,273,275,288,289,320,336,343,351,360
%N 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).
%C Theorem (an analogue of Korselt's criterion).
%C For a composite number n the following statements are equivalent:
%C (1) n is a 2-Carmichael number,
%C (2) for any prime divisor p of n, (p-1)*(p+1) | n*(n-1)*(n+1).
%H Amiram Eldar, <a href="/A336663/b336663.txt">Table of n, a(n) for n = 1..10000</a>
%H Eugene Karolinsky and Dmytro Seliutin, <a href="https://arxiv.org/abs/2001.10315">Carmichael numbers for GL(m)</a>, arXiv:2001.10315 [math.NT], 2020.
%t 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 *)
%o (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
%Y Cf. A002997.
%K nonn
%O 1,1
%A _Dmytro Seliutin_, Jul 29 2020
%E More terms from _Jinyuan Wang_, Jul 29 2020