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%I #7 Nov 08 2013 10:28:51
%S 1,2,1,5,3,1,2,1,23,2,1,27,3,1,2,1,4,2,1,3,2,7,1,2,3,1,5,4,2,1,6,3,2,
%T 1,12,2,1,3,4,1,2,5,3,1,2,4,3,1,2,1,6,2,3,21,4,7,5,8,1,2,73,3,1,2,4,3,
%U 26,1,2,5,1,3,9,10,2,4,6,20,1,3,2,11,1,4,2,5,3,7,1,2,3,4,1,6,29,2,3,5,8,1
%N Bases m in solutions of the Thue equation s^2 = m^5+z, sorted along increasing z.
%C The equation has solutions for the positive z listed in A152412.
%C A177761 and this sequence here show pairs (s,m) that solve given these z>0. (The case z=0 has infinitely many solutions which are not included here.)
%C There is no 1-to-1 relation to these z because more than one (s,m) may exist for some z, in case of which all are listed here.
%F A177761(n)^2 = a(n)^5 + A152412(k) for some k>1.
%e (s=59, m=5=a(57), z=356) and (s=182, m=8=a(58), z=356) are solutions associated with z = A152412(57) =356.
%e (s=20, m=2=a(60), z=368) and (s=45531, m=73=a(61), z=368) are solutions associated with z = A152412(59) =368.
%t aa = {}; bb = {}; cc = {}; Do[Do[If[(N[Sqrt[x^5 + n], 300] - Round[Sqrt[x^5 + n]])^2 < 10^-300, AppendTo[aa, n]; AppendTo[bb, x]; AppendTo[cc, Round[Sqrt[x^5 + n]]] , {x, 1, 100}], {n, 1, 100000}]; bb (*Artur Jasinski*)
%Y Cf. A152412, A177761.
%K nonn
%O 1,2
%A _Artur Jasinski_, May 13 2010
%E Examples and comment on coverage of multiple solutions added - _R. J. Mathar_, Aug 08 2010
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