Numbers n such that for every k the following condition holds: (*) The number of nonnegative solutions of the Diophantine equation x^2 + n*y^2 = 1+n*k^2 equals one half of the number of divisors of 1+n*k^2 if 1+n*k^2 is not a square and one half of 1 + the number of divisors of 1+n*k^2 if 1+n*k^2 is a square.
