

A292368


Numbers n with record number of primes p such that n*p is a LucasCarmichael number.


0



1, 21, 55, 385, 49105, 136081, 701569, 2830465, 7996801, 29158921, 49268737, 52617601
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OFFSET

1,2


COMMENTS

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).
The number of solutions is 0, 1, 2, 4, 5, 6, 7, 8, 10, 13, 15, 32.


LINKS



EXAMPLE

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.


MATHEMATICA

lucasCarmichaelQ[n_]:=!PrimeQ[n] && Union[Transpose[FactorInteger[n]][[2]]] == {1} && Union[Mod[n + 1, Transpose[FactorInteger[n]][[1]]+1]]=={0};
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 = {};
Do[m = numSol[n]; If[m > numSolmax, AppendTo[seq, n]; AppendTo[nums, m]; Print[{n, m}]; numSolmax = m], {n, 1, 100000}]; seq


CROSSREFS



KEYWORD

nonn,more


AUTHOR



STATUS

approved



