A287862 -id:A287862 - cp's OEIS frontend

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

Amiram Eldar, Sep 14 2017

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, ...

			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.

Cf. A006972, A287862, A292352.

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

cp's OEIS Frontend

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.

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