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%I #24 Mar 18 2022 00:10:51
%S 2,1729,15170835645,1801049058342701083
%N Cubefree taxicab numbers: the smallest cubefree number that is the sum of 2 positive cubes in n ways.
%C A necessary condition for the sum to be cubefree is that each pair of cubes be relatively prime.
%C 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
%H Bernd C. Kellner, <a href="https://arxiv.org/abs/1902.11283">On primary Carmichael numbers</a>, arXiv:1902.11283 [math.NT], 2019. See also <a href="http://math.colgate.edu/~integers/w38/w38.pdf">Integers</a> (2022) Vol. 22, #A38.
%H J. H. Silverman, <a href="http://www.maa.org/sites/default/files/images/upload_library/22/Ford/silverman331-340.pdf">Taxicabs and Sums of Two Cubes</a>, Amer. Math. Monthly, 100 (1993), 331-340.
%F a(n) >= A011541(n) for n > 0, with equality for n = 1, 2 (only?). - _Jonathan Sondow_, Oct 25 2013
%e 2 = 1^3 + 1^3,
%e 1729 = 12^3 + 1^3 = 10^3 + 9^3,
%e 15170835645 = 2468^3 + 517^3 = 2456^3 + 709^3 = 2152^3 + 1733^3,
%e 1801049058342701083 = 1216500^3 + 92227^3 = 1216102^3 + 136635^3 = 1207602^3 + 341995^3 = 1165884^3 + 600259^3.
%Y Cf. A011541.
%K hard,more,nonn
%O 1,1
%A Stuart Gascoigne (Stuart.G(AT)scoigne.com), Feb 28 2003
%E Name clarified by _Jeppe Stig Nielsen_, Aug 21 2020
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