%I A145996
%S A145996 0,1,2,4,243
%N A145996 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 A145996 When k = 1 the polynomial degenerates to degree 1.
%C A145996 Conjecture: This sequence is finite and complete.
%C A145996 This sequence is not the same as A005275 because 198815685282 does not 
               belong to this sequence.
%C A145996 No more values of k less than 2*10^7.
%C A145996 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*)
%C A145996 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]
%C A145996 = Sqrt[k/(k - 1)] Cos[3/5 ArcSin[1/Sqrt[k]]] [From Artur Jasinski (grafix(AT)csl.pl), 
               Oct 29 2008]
%t A145996 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*)
%Y A145996 A146160 [From Artur Jasinski (grafix(AT)csl.pl), Oct 29 2008]
%Y A145996 Sequence in context: A018742 A018747 A110067 this_sequence A005275 A009673 
               A018770
%Y A145996 Adjacent sequences: A145993 A145994 A145995 this_sequence A145997 A145998 
               A145999
%K A145996 nonn
%O A145996 1,3
%A A145996 Artur Jasinski (grafix(AT)csl.pl), Oct 26 2008