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%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
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