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%I #4 Mar 30 2012 17:22:33
%S 1,0,2,0,0,5,1,0,0,2,1,1,0,2,2,1,2,2,1,0,0,3,0,1,2,1,6,0,0,2,3,0,0,1,
%T 0,0,0,3,1,0,0,0,1,2,0,5,2,2,0,0,0,2,1,2,2,0,0,0,4,1,0,3,2,1,0,1,0,0,
%U 0,3,2,0,2,0,2,1,0,2,1,2,0,1,0,0,0,2,0,1,0,0,2,0,0,2,1,1,0,1,4
%N Number of solutions to the Lebesgue-Nagell equation x^2 + n = y^k with k > 2 and unique x.
%C Solutions such as 181^2+7 = 32^2 = 8^5 = 2^15 are counted only once. A094596 counts this as three solutions. Bugeaud, Mignotte and Siksek find all solutions for n <= 100.
%H Yann Bugeaud, Maurice Mignotte and Samir Siksek, <a href="http://www.arXiv.org/abs/math.NT/0405220">Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation</a>
%e a(4) = 2 because there are two solutions: 2^2+4=2^3 and 11^2+4=5^3.
%t Table[cnt=0; xLst={}; Do[x=Sqrt[y^k-n]; If[IntegerQ[x] && !MemberQ[xLst, x], cnt++; AppendTo[xLst, x]], {k, 3, 20}, {y, 600}]; cnt, {n, 2, 100}]
%Y Cf. A094596, A094598, A094599.
%K hard,nonn
%O 2,3
%A _T. D. Noe_, May 13 2004
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