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A292353 Numbers n with a record number of Lucas-Carmichael numbers that can be generated from them using an adjusted version of Erdős's method. 0
24, 72, 216, 240, 360, 720, 1440, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160 (list; graph; refs; listen; history; text; internal format)
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 number of generated Lucas-Carmichael numbers are 1, 3, 5, 9, 21, 169, 681, 900, 1842, 7250, 29132, 77482, 932187, 4970111, 7456418.

LINKS

Table of n, a(n) for n=1..15.

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++], {j, 1, Length[ps]}]; If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

CROSSREFS

Cf. A006972, A287861, A292352.

Sequence in context: A064200 A305065 A192833 * A189540 A329077 A042128

Adjacent sequences:  A292350 A292351 A292352 * A292354 A292355 A292356

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Sep 14 2017

STATUS

approved

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Last modified August 9 04:21 EDT 2020. Contains 336319 sequences. (Running on oeis4.)