A292354 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.

24, 48, 60, 144, 168, 240, 360, 720, 1440, 2520, 4320, 5040, 7560, 10080, 15120, 20160

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.

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

Links

Table of n, a(n) for n=1..16.

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 = {}; 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

Crossrefs

Cf. A006972, A287862, A292352.

Sequence in context: A341296 A338853 A343797 * A257861 A050497 A162282

Adjacent sequences: A292351 A292352 A292353 * A292355 A292356 A292357

Keyword

nonn,more

Author

Amiram Eldar, Sep 14 2017

Status

approved