| 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] (*Artur Jasinski*)
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. [From Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008]
= Sqrt[k/(k - 1)] Cos[3/5 ArcSin[1/Sqrt[k]]] [From Artur Jasinski (grafix(AT)csl.pl), Oct 29 2008]
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