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%I #35 Apr 24 2024 02:41:46
%S 29341,1152271,5481451,14913991,15247621,36765901,133800661,178482151,
%T 299736181,579606301,652969351,702683101,739444021,743404663,
%U 775368901,3215031751,4340265931,5871134179,8657319259,9293756581,12191597551,13734086221,14386156093,19331388805
%N Odd composite numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).
%C Carmichael numbers k such that A111076(k)^((k-1)/2) == -1 (mod k).
%C Note that if p is an odd prime, then A111076(p)^((p-1)/2) == -1 (mod p).
%C _Max Alekseyev_ proved (in a letter to the second author) that all these numbers have an odd number of prime factors, showing that if k is a term, then k is a Carmichael number m such that p-1 does not divide (m-1)/2 for every prime p|m (the numbers m form the supersequence A329799).
%C There are 6469 terms k of this sequence below 2^64:
%C 4240 with 3 prime factors, least is 29341 = 13*37*61,
%C 1790 with 5 prime factors, least is 4340265931 = 19*43*107*131*379,
%C 437 with 7 prime factors, least is 37038179683765 = 5*13*29*37*317*757*2213,
%C 2 with 9 prime factors, least is 1025735495681200591 = 7*19*31*67*79*163*199*271*5347.
%H Amiram Eldar, <a href="/A329538/b329538.txt">Table of n, a(n) for n = 1..10000</a> (calculated using data from Claude Goutier)
%H Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.
%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%t f[1, lam_] = 1; f[n_, lam_] := If[n < 5, n - 1, Module[{k = 1}, While[GCD[k, n] > 1 || MultiplicativeOrder[k, n] < lam, k++]; k]]; aQ[n_] := CompositeQ[n] && Divisible[n - 1 , (lam = CarmichaelLambda[n])] && PowerMod[f[n, lam], (n - 1)/2, n] == n - 1; Select[Range[1, 6*10^6, 2], aQ] (* after the _Charles R Greathouse IV_ at A111076 *)
%Y Cf. A002997, A111076, A329799.
%Y Subsequence of A262043.
%K nonn
%O 1,1
%A _Amiram Eldar_ and _Thomas Ordowski_, Nov 16 2019
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