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%I #9 Nov 06 2013 14:57:01
%S 0,1,2,4,243
%N Numbers k such that 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 has a rational root.
%C When k = 1 the polynomial degenerates to degree 1.
%C Conjecture: This sequence is finite and complete.
%C This sequence is not the same as A005275 because 198815685282 does not belong to this sequence.
%C No more values of k less than 2*10^7.
%C 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).
%C 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]
%C = sqrt(k/(k-1)) cos(3/5 arcsin(1/sqrt(k))). [_Artur Jasinski_, Oct 29 2008]
%t 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
%Y Cf. A146160.
%K nonn
%O 1,3
%A _Artur Jasinski_, Oct 26 2008
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