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

Author

Amiram Eldar, Sep 01 2017

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A002997, A287840.

Programs

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