A080642 Cubefree taxicab numbers: the smallest cubefree number that is the sum of 2 positive cubes in n ways.

2, 1729, 15170835645, 1801049058342701083

Offset: 1

Author

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

Keywords

• hard
• more
• nonn

Comments

A necessary condition for the sum to be cubefree 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

Examples

			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.
		

Links

• Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
• J. H. Silverman, Taxicabs and Sums of Two Cubes, Amer. Math. Monthly, 100 (1993), 331-340.

Crossrefs

Cf. A011541.

Formula

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

Extensions

Name clarified by Jeppe Stig Nielsen, Aug 21 2020