

A020898


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


11



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
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OFFSET

1,1


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


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), 220226.
Steven R. Finch, On a Generalized FermatWiles Equation [broken link]
Steven R. Finch, On a Generalized FermatWiles Equation [From the Wayback Machine]


EXAMPLE

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.


MATHEMATICA

(* 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 *)


CROSSREFS

Cf. A159843, A166246, A228499, A254324, A254326.
Sequence in context: A061416 A190247 A020897 * A184779 A200926 A047277
Adjacent sequences: A020895 A020896 A020897 * A020899 A020900 A020901


KEYWORD

nonn,nice,changed


AUTHOR

Steven Finch


EXTENSIONS

Entry revised by N. J. A. Sloane, Aug 12 2004
Links updated by Max Alekseyev, Oct 17 2007 and Dec 12 2007


STATUS

approved



