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%I #25 Mar 26 2022 17:46:04
%S 9,7,2,812918,18,217,4,3,9730705,332,14,135,3,19,156,16,15584139827,3,
%T 139643,6,1541,4,2220422932,5,14,4,445,12205,9,8,16234,815,31,4
%N Consider positive integer solutions to x^3+ y^3 = z^3 - n or 'Fermat near misses' of 1, 2, 3 ... Arrange known solutions by increasing values of n. Sequence gives value of lowest z for a given n.
%C The submitted values are for z when 0 < n < 51. There is no solution for any n congruent to 4 or 5 mod 9. This eliminates 4,5,13,14,22,23,31,32,40,41,49 and 50 in the 0-to-50 range.
%C Per the Elsenhans and Jahnel link there are no solutions found for 3, 33, 39 and 42 in the 0-to-50 range, with a search bound of 10^14.
%C If sequences could contain 'nil' for no solution, and '?' for cases where a solution is not known, but might exist, then a more concise definition is possible: Least positive integer such that a(n)^3 - n is the sum of two positive cubes. The sequence would then start: 9, 7, ?, nil, nil, 2.
%H Andreas-Stephan Elsenhans and Joerg Jahnel, <a href="http://www.uni-math.gwdg.de/jahnel/Arbeiten/Liste/threecubes_20070419.txt">List of solutions of x^3 + y^3 + z^3 = n for n < 1000 neither a cube nor twice a cube</a>
%H D.R. Heath-Brown, W.M. Lioen and H.J.J. te Riele, <a href="http://euler.free.fr/docs/HLR93.pdf"> on Solving the Diophantine Equation x^3 + y^3 + z^3 = k on a Vector Computer</a>
%H Kenji Koyama, Yukio Tsuruoka, and Hiroshi Sekigawa, <a href="http://dx.doi.org/10.1090/S0025-5718-97-00830-2">On Searching For Solutions of the Diophantine Equation x^3 + y^3 + z^3 = n</a>, Math. Comp. 66 (1997), 841-851.
%H Eric S. Rowland, <a href="https://ericrowland.github.io/papers/Known_families_of_integer_solutions_of_x^3+y^3+z^3=n.pdf">Known Families of Integer Solutions of x^3 + y^3 + z^3 = n</a>
%F Author conjectures that no explicit formula or recurrence exists.
%e 6^3 + 8^3 = 9^3 - 1: There are no solutions when n = 1 for z < 9, thus the first term is 9.
%e 5^3 + 6^3 = 7^3 - 2: There are no solutions for z < 7, thus the second term is 7.
%e It is unknown if there is a solution when n = 3.
%e It is known there are no solutions when n = 4 and 5.
%e 1^3 + 1^3 = 2^3 - 6, thus the third term is 2.
%o (Ruby)
%o # x^3 + y^3 = z^3 - n
%o # Solve for all z less than z_limit, and
%o # n less than n_limit.
%o # When n = 7, z = 812918 and faster code and language are needed.
%o # However, by optimizing this code slightly and running for 2 days
%o # the author was able to search all z < 164000 and n < 100
%o #
%o n_limit = 7 # Configure as desired
%o z_limit = 20 # Configure as desired
%o h = {}
%o (2..z_limit).each{ |z|
%o . (1..(z-1)).each{ |y|
%o . (1..(y)).each{ |x|
%o . n = z*z*z - x*x*x - y*y*y
%o . if n > 0 && n < n_limit && h[n].nil?
%o . puts "Found z = #{z} when #{x}^^3 + #{y}^^3 = #{z}^^3 - #{n}"
%o . h[n] = z
%o . end
%o } } } print "\nPartial sequence generated when n < #{n_limit} and z is searched to #{z_limit} is:\n"
%o h.sort.each{|k,v| print "#{v}, " }
%o print "\b\b \n"
%Y Cf. A050788, A159935.
%K nonn
%O 1,1
%A _Andy Martin_, Feb 20 2010
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