login
A292352
Numbers that generate Lucas-Carmichael numbers using an adjusted version of Erdős's method.
2
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
1,1
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, ...
EXAMPLE
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.
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
CROSSREFS
Sequence in context: A091192 A067807 A224907 * A307342 A067341 A307682
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 14 2017
STATUS
approved