A292354 - OEIS

%I #5 Sep 17 2017 22:50:12

%S 24,48,60,144,168,240,360,720,1440,2520,4320,5040,7560,10080,15120,

%T 20160

%N Numbers n with a record size of the largest Lucas-Carmichael number that can be generated from them 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 Lucas-Carmichael 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 Lucas-Carmichael number.

%C The corresponding largest Lucas-Carmichael numbers are 8855, 18095, 357599, 1010735, 406335215, 1087044101759, 4467427448759, ...

%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 Lucas-Carmichael numbers thus a(1)=24.

%t a = {}; cmax = 0; 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 = Max[c, c1]], {j, 1, Length[ps]}]; If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

%Y Cf. A006972, A287862, A292352.

%K nonn,more

%O 1,1

%A _Amiram Eldar_, Sep 14 2017