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
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
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Sep 01 2017
STATUS
approved