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%I #37 Jan 24 2023 09:45:45
%S 2,6,7,9,12,13,15,16,17,19,20,22,26,28,30,31,33,34,35,37,42,43,48,49,
%T 50,51,53,54,56,58,61,62,63,65,67,68,69,70,71,72,75,78,79,84,85,86,87,
%U 89,90,91,92,94,96,97,98,103,104,105,106,107,110,114,115,117,120
%N Sum of two nonzero rational cubes.
%C n such that x^3 + y^3 = n*z^3 has a solution in nonnegative integers x,y,z.
%C Dolan on page 37 states: "Equation 1 is insoluble if n = 31, 38, 67, 76 or 95." - _Michael Somos_, Nov 18 2021
%C See the Selmer 1951 article pp. 357-360, Table 6, "The number g of generators and the basic solutions of the equation X^3 + Y^3 = AZ^3, A cubefree and <= 500. - _Michael Somos_, Feb 15 2022
%H S. W. Dolan, <a href="https://www.jstor.org/stable/3617304">On expressing numbers as the sum of two cubes</a>, The Mathematical Gazette, Vol. 66, No. 436 (Mar., 1982), pp. 31-38. See table of solutions on page 32. - _Michael Somos_, Nov 18 2021
%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]
%H Ernst S. Selmer, <a href="http://dx.doi.org/10.1007/BF02395746">The diophantine equation ax^3 + by^3 + cz^3 = 0</a>, Acta Math. 85 (1951), pp. 203-362.
%e 6*21^3 = 37^3 + 17^3, 7*3^3 = 5^3 + 4^3, 9*1^3 = 2^3 + 1^3, 12*39^3 = 89^3 + 19^3, 13*3^3 = 7^3 + 2^3, 15*294^3 = 683^3 + 397^3, ... - _Michael Somos_, Nov 18 2021
%e 31*42^3 = 137^3 + (-65)^3, 67*1323^3 = 5353^3 + 1208^3. - _Michael Somos_, Feb 12 2022
%t (* A naive program with a few pre-computed terms *) nmax = 120; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Reap[Do[n = CubeFreePart[x*y*(x+y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union; A020897 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* _Jean-François Alcover_, Apr 02 2012 *)
%Y Subsequence of A159843.
%K nonn,nice
%O 1,1
%A _Steven Finch_
%E Offset corrected by _Arkadiusz Wesolowski_, Aug 15 2013
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