%I
%S 2,6,7,9,12,13,15,17,19,20,22,26,28,30,31,33,34,35,37,42,43,49,50,51,
%T 53,58,61,62,63,65,67,68,69,70,71,75,78,79,84,85,86,87,89,90,91,92,94,
%U 97,98,103,105,106,107,110,114,115,117,123,124,126,127,130
%N Positive cubefree integers n such that the Diophantine equation X^3 + Y^3 = n*Z^3 has solutions.
%C These numbers are the cubefree sums of two nonzero rational cubes.
%C This sequence does not contain A202679, which has members that are not cubefree.  _Robert Israel_, Mar 16 2016
%D B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Amer. Math. Soc., 1964.
%D L. E. Dickson, History of The Theory of Numbers, Vol. 2, Chap. XXI, Chelsea NY 1966.
%D L. J. Mordell, Diophantine Equations, Ac. Press, Chap. 15.
%H David W. Wilson, <a href="/A020898/b020898.txt">Table of n, a(n) for n = 1..255</a> (from Finch paper)
%H J. H. E. Cohn, <a href="http://www.jstor.org/stable/2691526">The £ 450 question</a>, Math. Mag., 73 (no. 3, 2000), 220226.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a Generalized FermatWiles 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 FermatWiles Equation</a> [From the Wayback Machine]
%e 37^3 + 17^3 = 6*21^3 is the smallest positive solution for n = 6 (found by Lagrange).
%e 5^3 + 4^3 = 7*3^3 is the smallest positive solution for n = 7.
%t (* A naive program with a few precomputed terms *) nmax = 130; 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; A020898 = Union[nn, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107, 123}](* _JeanFrançois Alcover_, Mar 30 2012 *)
%Y Cf. A159843, A166246, A254324, A254326.
%K nonn,nice
%O 1,1
%A _Steven Finch_
%E Entry revised by _N. J. A. Sloane_, Aug 12 2004
%E Links updated by _Max Alekseyev_, Oct 17 2007 and Dec 12 2007
