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A293203 Numbers k such that A090086(k), the smallest pseudoprime to base k (not necessarily exceeding k), is a Carmichael number.

%I A293203 #29 Jun 09 2020 03:31:42
%S A293203 700,1040,1150,1848,2590,2660,6710,6862,7000,7716,7852,8060,8528,9275,
%T A293203 9875,10103,10640,11830,12010,12688,13340,16520,17350,17570,17960,
%U A293203 18130,18340,19203,19272,19420,19820,19978,20410,20442,20480,20612,20720,23016,23463
%N A293203 Numbers k such that A090086(k), the smallest pseudoprime to base k (not necessarily exceeding k), is a Carmichael number.
%C A293203 The corresponding Carmichael numbers are 561, 561, 561, 1105, 561, 561, 1729, 561, 561, 1105, 561, 561, 561, 561, 561, 561, 561, 561, 561, ...
%C A293203 Andrzej Schinzel proved that this sequence is infinite. Conjecture: if A090086(n) is a Carmichael number k, then k < n. - _Thomas Ordowski_, Aug 08 2018
%H A293203 Amiram Eldar, <a href="/A293203/b293203.txt">Table of n, a(n) for n = 1..10000</a>
%H A293203 Andrzej Rotkiewicz, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa91/aa9114.pdf">Periodic sequences of pseudoprimes connected with Carmichael number and the least period of the function l_x^C</a>, Acta Arithmetica, Vol. 91, No. 1 (1999), pp. 75-83.
%H A293203 Andrzej Rotkiewicz, <a href="https://doi.org/10.1007/978-94-011-4271-7_28">Solved and unsolved problems on pseudoprime numbers and their generalizations</a>, Applications of Fibonacci numbers, Springer Netherlands, 1999, pp. 293-306.
%e A293203 700 is the sequence since A090086(700) = 561 is a Carmichael number.
%t A293203 carmichaelQ[n_] := Divisible[n - 1, CarmichaelLambda[n]] && ! PrimeQ[n];
%t A293203 f[n_] := Block[{k = 1}, While[GCD[n, k] > 1 || PrimeQ[k] || PowerMod[n, k - 1, k] != 1, j = k++]; k]; Select[Range[10000], carmichaelQ[f[#]] &] (* after Robert G.Wilson v at A090086 *)
%Y A293203 Cf. A002997, A090086.
%K A293203 nonn
%O A293203 1,1
%A A293203 _Amiram Eldar_, Oct 12 2017