%I
%S 1,21,55,385,49105,136081,701569,2830465,7996801,29158921,49268737,
%T 52617601
%N Numbers n with record number of primes p such that n*p is a LucasCarmichael number.
%C Given a number n and a prime number p such that n*p is a LucasCarmichael number, then (p+1)(n1), so the number of prime solution p given n is bounded by the number of divisors of (n1).
%C The number of solutions is 0, 1, 2, 4, 5, 6, 7, 8, 10, 13, 15, 32.
%e 21 has one prime number, 19, such that 21*19 = 399 is a LucasCarmichael number. 55 has 2 prime numbers, 17 and 53, such that 55*17 = 935 and 55*53 = 2915 are LucasCarmichael numbers.
%t lucasCarmichaelQ[n_]:=!PrimeQ[n] && Union[Transpose[FactorInteger[n]][[2]]] == {1} && Union[Mod[n + 1, Transpose[FactorInteger[n]][[1]]+1]]=={0};
%t numSol[n_]:=Module[{m = 0}, ds = Divisors[n1]; Do[p = ds[[k]]1; If[!PrimeQ[p], Continue[]]; If[! lucasCarmichaelQ[p*n], Continue[]]; m++, {k, 1, Length[ds]}]; m]; numSolmax = 1; seq = {}; nums = {};
%t Do[m = numSol[n]; If[m > numSolmax, AppendTo[seq, n]; AppendTo[nums, m]; Print[{n, m}]; numSolmax = m], {n, 1, 100000}]; seq
%Y Cf. A006972.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Sep 15 2017
