%I
%S 24,36,40,48,60,72,80,84,96,108,120,144,168,180,192,200,216,240,252,
%T 270,300,324,336,360,384,400,420,432,440,468,480,504,528,540,576,588,
%U 600,624,648,660,672,714,720,744,756,768,792,810,840,864,900,912,936,960
%N Numbers that generate LucasCarmichael numbers using an adjusted version of Erdős's method.
%C Erdős showed in 1956 how to construct Carmichael numbers from a given number n (see A287840). With appropriate sign changes the method can be used to generate LucasCarmichael numbers. Given a number n, let P be the set of primes p such that (p+1)n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == 1 (mod n) then c is a LucasCarmichael number.
%C Numbers with only one generated LucasCarmichael number: 24, 36, 40, 48, 60, 80, 84, 96, 108, 200, 252, 270, 300, 324, 336, 400, 440, 468, ...
%e The set of primes for n = 24 is P={2, 3, 5, 7, 11, 23}. One subset, {5, 7, 11, 23} have c == 1 (mod n): c = 5*7*11*23 = 8855. 24 is the least number that generates LucasCarmichael numbers thus a(1)=24.
%t a = {}; Do[p = Select[Divisors[n]  1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c++], {j, 1, Length[ps]}]; If[c > 0, AppendTo[a, n]], {n, 1, 1000}]; a
%Y Cf. A006972, A287840.
%K nonn
%O 1,1
%A _Amiram Eldar_, Sep 14 2017
