Cabtaxi Number

The th cabtaxi number is the smallest positive integer that can be written in distinct ways as a sum of two (not necessarily positive) integer cubes, where is a positive integer and interchanging the two summands does not give a new representation. The name comes from the taxicab number, which is the smallest number representable in ways as a sum of positive cubes. The first few are 1, 91, 728, 2741256, 6017193, 1412774811, 11302198488, 137513849003496, 424910390480793000, 933528127886302221000, ... (OEIS A047696), as listed below. For , these minima were computed by Bernstein (2001).

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The ninth term was found by Duncan Moore (2005). Christian Boyer found the tenth term as an upper bound in 2006 (Boyer 2008), and Hollerbach (2008) subsequently reported verifying its minimality.