

A292352


Numbers that generate LucasCarmichael numbers using an adjusted version of ErdÅ‘s'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
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OFFSET

1,1


COMMENTS

ErdÅ‘s 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.
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, ...


LINKS

Table of n, a(n) for n=1..54.


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 LucasCarmichael 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

Cf. A006972, A287840.
Sequence in context: A091192 A067807 A224907 * A307342 A067341 A307682
Adjacent sequences: A292349 A292350 A292351 * A292353 A292354 A292355


KEYWORD

nonn


AUTHOR

Amiram Eldar, Sep 14 2017


STATUS

approved



