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%I #13 Feb 04 2026 08:37:24
%S 561,1105,2465,6601,8911,29341,41041,75361,162401,278545,530881,
%T 1050985,2100901,4335241,8719309,16778881,33596641,67371265,134809921,
%U 270857521,540066241,1074363265,2159003281,4295605861,8612234401,17190510961,34364331001,68910004801,137691502081,274895715601,549813672001,1099910311201
%N Smallest n-bit Carmichael number, or 0 if no such number exists.
%C a(n) > 0 for large enough n, see Larsen link. Probably a(n) > 0 for all n >= 10.
%H Amiram Eldar, <a href="/A375875/b375875.txt">Table of n, a(n) for n = 10..74</a>
%H Daniel Larsen, <a href="https://doi.org/10.1093/imrn/rnac203">Bertrand's Postulate for Carmichael Numbers</a>, International Mathematics Research Notices, Vol. 2023, No. 15 (2023), pp. 13072-13098; <a href="https://arxiv.org/abs/2111.06963">arXiv preprint</a>, arXiv:2111.06963 [math.NT], 2021-2023.
%F a(n) ~ 2^n due to Larsen.
%t carmQ[k_] := CompositeQ[k] && Divisible[k-1, CarmichaelLambda[k]]; a[n_] := Module[{k = 2^(n-1)}, While[!carmQ[k], k++]; If[k > 2^n, 0, k]]; Array[a, 10, 10] (* _Amiram Eldar_, Feb 04 2026 *)
%o (PARI) a(n)=forsquarefree(t=2^(n-1),2^n, my(f=t[2]); if(#f~>1 && f[1,1]>2 && Korselt(t[1],f), return(t[1])))
%Y Conjecturally a subsequence of A002997.
%K nonn
%O 10,1
%A _Charles R Greathouse IV_, Sep 01 2024