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%I #26 Feb 12 2021 06:24:07
%S 2,31,33,64,211,242,244,275,486,781,992,1023,1025,1056,1267,2048,2101,
%T 2882,3093,3124,3126,3157,3368,4149,4651,6250,6752,7533,7744,7775,
%U 7777,7808,8019,8800,9031,10901,13682,15552,15783,15961,16564
%N Positive numbers k such that k = x^5 + y^5 has a solution in nonzero integers x, y.
%C 68101 = (15/2)^5 + (17/2)^5 is believed to be the smallest positive integer k which is the sum of two nonzero fifth powers of rational numbers but not the sum of two nonzero fifth powers of integers.
%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.
%H Vincenzo Librandi, <a href="/A020896/b020896.txt">Table of n, a(n) for n = 0..5000</a>
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a Generalized Fermat-Wiles Equation</a> [broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On a Generalized Fermat-Wiles Equation</a> [From the Wayback Machine]
%F See Theorem 3.5.6 of J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.
%e 31 = 2^5 + (-1)^5.
%t Select[Union[Total/@(Select[Tuples[Range[-8,8],{2}], !MemberQ[#, 0]&]^5)],#>0&] (* _Harvey P. Dale_, Apr 03 2011 *)
%Y Cf. A001481, A020897, A003336.
%K nonn,nice
%O 0,1
%A _Steven Finch_
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