A287861 Numbers n with a record number of Carmichael numbers that can be generated from them using Erdős's method.

36, 120, 180, 240, 360, 540, 720, 1080, 1200, 1260, 1680, 2160, 2520, 3780, 5040, 7560, 10080, 15120, 25200

Offset

1,1

Comments

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (typically with many divisors). 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 Carmichael number.

The corresponding number of generated Carmichael numbers are 2, 3, 4, 8, 11, 16, 26, 30, 36, 57, 79, 204, 466, 610, 7253, 9778, 58058, 1244090, 5963529.

Links

Table of n, a(n) for n=1..19.

Paul Erdős, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), pp. 201-206.

Andrew Granville, Primality testing and Carmichael numbers, Notices of the American Mathematical Society, Vol. 39 No. 6 (1992), pp. 696-700.

Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

Example

The set of primes for n = 36 is P={5, 7, 13, 19, 37}. Two subsets, {7, 13, 19} and {7, 13, 19, 37} have c == 1 (mod n): c = 7*13*19 = 1729 and c = 7*13*19*37 = 63973. 36 is the first number that generates Carmichael numbers thus a(1)=36.

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. A002997, A287840.

Sequence in context: A044668 A129367 A350500 * A242356 A165966 A307939

Adjacent sequences: A287858 A287859 A287860 * A287862 A287863 A287864

Keyword

nonn,more

Author

Amiram Eldar, Sep 01 2017

Status

approved