A292352 Numbers that generate Lucas-Carmichael numbers using an adjusted version of Erdős's method.

24, 36, 40, 48, 60, 72, 80, 84, 96, 108, 120, 144, 168, 180, 192, 200, 216, 240, 252, 270, 300, 324, 336, 360, 384, 400, 420, 432, 440, 468, 480, 504, 528, 540, 576, 588, 600, 624, 648, 660, 672, 714, 720, 744, 756, 768, 792, 810, 840, 864, 900, 912, 936, 960

Offset: 1

Author

Amiram Eldar, Sep 14 2017

Keywords

• nonn

Comments

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.

Numbers with only one generated Lucas-Carmichael number: 24, 36, 40, 48, 60, 80, 84, 96, 108, 200, 252, 270, 300, 324, 336, 400, 440, 468, ...

Examples

			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.
		

Crossrefs

Cf. A006972, A287840.

Programs

Mathematica

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