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A080642 Cube-free taxicab numbers: the smallest cube-free number that is the sum of 2 cubes in n ways. 2
2, 1729, 15170835645, 1801049058342701083 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A necessary condition for the sum to be cube-free is that each pair of cubes be relatively prime.

If the sequence is infinite, then the Mordell-Weil rank of the elliptic curve of rational solutions to x^3 + y^3 = a(n) tends to infinity with n. In fact, the rank exceeds C*log(n) for some constant C>0 (see Silverman p. 339). - Jonathan Sondow, Oct 22 2013

LINKS

Table of n, a(n) for n=1..4.

J. H. Silverman, Taxicabs and Sums of Two Cubes, Amer. Math. Monthly, 100 (1993), 331-340.

FORMULA

a(n) >= A011541(n) for n > 0, with equality for n = 1, 2 (only?). - Jonathan Sondow, Oct 25 2013

EXAMPLE

2 = 1^3 + 1^3

1729 = 12^3 + 1^3 = 10^3 + 9^3

15170835645 = 2468^3 + 517^3 = 2456^3 + 709^3 = 2152^3 + 1733^3

1801049058342701083 = 1216500^3 + 92227^3 = 1216102^3 + 136635^3 = 1207602^3 + 341995^3 = 1165884^3 + 600259^3

CROSSREFS

Cf. A011541.

Sequence in context: A129061 A233132 A011541 * A108331 A162554 A167840

Adjacent sequences:  A080639 A080640 A080641 * A080643 A080644 A080645

KEYWORD

hard,more,nonn

AUTHOR

Stuart Gascoigne (Stuart.G(AT)scoigne.com), Feb 28 2003

STATUS

approved

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Last modified July 23 17:53 EDT 2014. Contains 244870 sequences.