When k = 1 the polynomial degenerates to degree 1.
Conjecture: This sequence is finite and complete.
This sequence is not the same as A005275 because 198815685282 does not belong to this sequence.
No more values of k less than 2*10^7.
One of the root of quintic polynomial 4 k  k^2 + 5 k^2 x + (20 k  20 k^2) x^3 + (16  32 k + 16 k^2) x^5 is Hypergeometric2F1(1/5,4/5,1/2,1/k).
Precisely for k belonging to this sequence, Hypergeometric2F1(1/5,4/5,1/2,1/k) is algebraic number of 4 degree, otherwise it is of degree 5. [Artur Jasinski, Oct 26 2008]
= sqrt(k/(k1)) cos(3/5 arcsin(1/sqrt(k))). [Artur Jasinski, Oct 29 2008]
