1,3

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]

a = {}; Do[If[Length[FactorList[(4 k - k^2 + 5 k^2 x + (20 k - 20 k^2) x^3 + (16 - 32 k + 16 k^2) x^5)]] > 2, AppendTo[a, k]; Print[k]], {k, 1, 20000000}]; a (*Artur Jasinski*)

A146160 [From Artur Jasinski (grafix(AT)csl.pl), Oct 29 2008]

Sequence in context: A018742 A018747 A110067 this_sequence A005275 A009673 A018770

Adjacent sequences: A145993 A145994 A145995 this_sequence A145997 A145998 A145999

nonn

Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008