A Curzon number is defined to be a positive integer n for which 2n + 1 is evenly divisible by 2 × n + 1.

Generalized Curzon numbers are those where the positive integer n, using a base integer k, satisfy the condition that kn + 1 is evenly divisible by k × n + 1.

Base here does not imply the radix of the counting system; rather the integer the equation is based on. All calculations should be done in base 10.

Generalized Curzon numbers only exist for even base integers.



Task 
 Find and show the first 50 Generalized Curzon numbers for even base integers from 2 through 10.


Stretch
 Find and show the one thousandth.


See also
 Numbers Aplenty - Curzon numbers
 OEIS:A224486 - Numbers k such that 2*k+1 divides 2^k+1 (Curzon numbers)
and even though it is not specifically mentioned that they are Curzon numbers:

 OEIS:A230076 - (A007521(n)-1)/4 (Generalized Curzon numbers with a base 4)




sub curzon ($base) { lazy (1..∞).hyper.map: { $_ if (exp($_, $base) + 1) %% ($base × $_ + 1) } };
 
for <2 4 6 8 10> {
    my $curzon = .&curzon;
    say "\nFirst 50 Curzon numbers using a base of $_:\n" ~
      $curzon[^50].batch(25)».fmt("%4s").join("\n") ~
      "\nOne thousandth: " ~ $curzon[999]
}