A020898 Positive cubefree integers n such that the Diophantine equation X^3 + Y^3 = n*Z^3 has solutions.

2, 6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 49, 50, 51, 53, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 97, 98, 103, 105, 106, 107, 110, 114, 115, 117, 123, 124, 126, 127, 130

Offset: 1

Author

Steven Finch

Keywords

• nonn
• nice

Comments

These numbers are the cubefree sums of two nonzero rational cubes.

This sequence does not contain A202679, which has members that are not cubefree. - Robert Israel, Mar 16 2016

Notice that 34^3 + 74^3 = 48*21^3 = 6*42^3 because 48 = 6*2^3 is not cubefree, but now 17^3 + 37^3 = 6*21^3 and 6 is already listed in the sequence. - Michael Somos, Mar 13 2023

Examples

			37^3 + 17^3 = 6*21^3 is the smallest positive solution for n = 6 (found by Lagrange).
5^3 + 4^3 = 7*3^3 is the smallest positive solution for n = 7.
		

References

• B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Amer. Math. Soc., 1964.
• L. E. Dickson, History of The Theory of Numbers, Vol. 2, Chap. XXI, Chelsea NY 1966.
• L. J. Mordell, Diophantine Equations, Academic Press, Chap. 15.

Links

• David W. Wilson, Table of n, a(n) for n = 1..255 (from Finch paper)
• J. H. E. Cohn, The £ 450 question, Math. Mag., 73 (no. 3, 2000), 220-226.
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]

Crossrefs

Cf. A159843, A166246, A228499, A254324, A254326.

Programs

Mathematica

(* A naive program with a few pre-computed 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}](* Jean-François Alcover, Mar 30 2012 *)

Extensions

Entry revised by N. J. A. Sloane, Aug 12 2004

Links updated by Max Alekseyev, Oct 17 2007 and Dec 12 2007