OFFSET
1,2
COMMENTS
If n*p is a Carmichael number, where p is a prime, then (p-1)|(n-1), so given n, the number of possible primes is bounded by the number of divisors of n-1.
The corresponding number of solutions is 0, 1, 2, 3, 5, 7, 10, 12, 14, 18, 26, 30, 33, 55.
EXAMPLE
33 has 1 prime number, 17, such that 33*17 = 561 is a Carmichael number.
85 has 2 prime numbers, 13 and 29, such that 85*13 = 1105 and 85*29 = 2465 are Carmichael numbers.
MATHEMATICA
carmichaelQ[n_]:= Not[PrimeQ[n]] && Divisible[n-1, CarmichaelLambda[n]];
numSol[n_] := Module[{m = 0}, ds = Divisors[n-1]; Do[p = ds[[k]] + 1; If[! PrimeQ[p], Continue[]]; If[! carmichaelQ[p*n], Continue[]]; m++, {k, 1, Length[ds] - 1}]; m]; numSolmax = -1; seq = {}; nums = {}; Do[m = numSol[n]; If[m > numSolmax, AppendTo[seq, n]; AppendTo[nums, m]; numSolmax = m], {n, 1, 10^8}]; seq
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Sep 15 2017
EXTENSIONS
a(11)-a(14) from Giovanni Resta, May 08 2020
STATUS
approved