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 A020897 Sum of two nonzero rational cubes. 8
 2, 6, 7, 9, 12, 13, 15, 16, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 54, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117, 120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS n such that x^3 + y^3 = n*z^3 has a solution in nonnegative integers x,y,z. Dolan on page 37 states: "Equation 1 is insoluble if n = 31, 38, 67, 76 or 95." - Michael Somos, Nov 18 2021 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 LINKS S. W. Dolan, On expressing numbers as the sum of two cubes, The Mathematical Gazette, Vol. 66, No. 436 (Mar., 1982), pp. 31-38. See table of solutions on page 32. - Michael Somos, Nov 18 2021 Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link] Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine] Ernst S. Selmer, The diophantine equation ax^3 + by^3 + cz^3 = 0, Acta Math. 85 (1951), pp. 203-362. EXAMPLE 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 31*42^3 = 137^3 + (-65)^3, 67*1323^3 = 5353^3 + 1208^3. - Michael Somos, Feb 12 2022 MATHEMATICA (* A naive program with a few pre-computed terms *) nmax = 120; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[], Mod[#[], 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 *) CROSSREFS Subsequence of A159843. Sequence in context: A344167 A061416 A190247 * A020898 A184779 A200926 Adjacent sequences: A020894 A020895 A020896 * A020898 A020899 A020900 KEYWORD nonn,nice AUTHOR EXTENSIONS Offset corrected by Arkadiusz Wesolowski, Aug 15 2013 STATUS approved

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Last modified March 29 19:01 EDT 2023. Contains 361599 sequences. (Running on oeis4.)