Cabtaxi number

In number theory, the $n$-th cabtaxi number, typically denoted $Cabtaxi(n)$, is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in $n$ ways. Such numbers exist for all $n$, which follows from the analogous result for taxicab numbers.

Known cabtaxi numbers
Only 10 cabtaxi numbers are known :

$$\begin{align} \mathrm{Cabtaxi}(1) =& \ 1 \\ &= 1^3 + 0^3 \\[6pt] \mathrm{Cabtaxi}(2) =& \ 91 \\ &= 3^3 + 4^3 \\ &= 6^3 - 5^3 \\[6pt] \mathrm{Cabtaxi}(3) =& \ 728 \\ &= 6^3 + 8^3 \\ &= 9^3 - 1^3 \\  &= 12^3 - 10^3 \\[6pt] \mathrm{Cabtaxi}(4) =& \ 2741256 \\ &= 108^3 + 114^3 \\ &= 140^3 - 14^3 \\  &= 168^3 - 126^3 \\  &= 207^3 - 183^3 \\[6pt] \mathrm{Cabtaxi}(5) =& \ 6017193 \\ &= 166^3 + 113^3 \\ &= 180^3 + 57^3 \\  &= 185^3 - 68^3 \\  &= 209^3 - 146^3 \\  &= 246^3 - 207^3 \\[6pt] \mathrm{Cabtaxi}(6) =& \ 1412774811 \\ &= 963^3 + 804^3 \\ &= 1134^3 - 357^3 \\  &= 1155^3 - 504^3 \\  &= 1246^3 - 805^3 \\  &= 2115^3 - 2004^3 \\  &= 4746^3 - 4725^3 \\[6pt] \mathrm{Cabtaxi}(7) =& \ 11302198488 \\ &= 1926^3 + 1608^3 \\ &= 1939^3 + 1589^3 \\  &= 2268^3 - 714^3 \\  &= 2310^3 - 1008^3 \\  &= 2492^3 - 1610^3 \\  &= 4230^3 - 4008^3 \\  &= 9492^3 - 9450^3 \\[6pt] \mathrm{Cabtaxi}(8) =& \ 137513849003496 \\ &= 22944^3 + 50058^3 \\ &= 36547^3 + 44597^3 \\  &= 36984^3 + 44298^3 \\  &= 52164^3 - 16422^3 \\  &= 53130^3 - 23184^3 \\  &= 57316^3 - 37030^3 \\  &= 97290^3 - 92184^3 \\   &= 218316^3 - 217350^3 \\[6pt] \mathrm{Cabtaxi}(9) =& \ 424910390480793000 \\ &= 645210^3 + 538680^3 \\  &= 649565^3 +  532315^3 \\  &= 752409^3 -  101409^3 \\  &= 759780^3 -  239190^3 \\  &= 773850^3 -  337680^3 \\  &= 834820^3 -  539350^3 \\  &= 1417050^3 - 1342680^3 \\  &= 3179820^3 - 3165750^3 \\  &= 5960010^3 - 5956020^3 \\[6pt] \mathrm{Cabtaxi}(10) =& \ 933528127886302221000 \\ &= 8387730^3 + 7002840^3 \\ &= 8444345^3 + 6920095^3 \\  &= 9773330^3 - 84560^3 \\  &= 9781317^3 - 1318317^3 \\  &= 9877140^3 - 3109470^3 \\  &= 10060050^3 - 4389840^3 \\  &= 10852660^3 - 7011550^3 \\  &= 18421650^3 - 17454840^3 \\  &= 41337660^3 - 41154750^3 \\  &= 77480130^3 - 77428260^3 \end{align}$$

History
Cabtaxi(2) was known to François Viète and Pietro Bongo in the late 16th century in the equivalent form $$3^3+4^3+5^3=6^3$$. The existence of Cabtaxi(3) was known to Leonhard Euler, but its actual solution was not found until later, by Edward B. Escott in 1902.

Cabtaxi(4) through and Cabtaxi(7) were found by Randall L. Rathbun in 1992; Cabtaxi(8) was found by Daniel J. Bernstein in 1998. Cabtaxi(9) was found by Duncan Moore in 2005, using Bernstein's method. Cabtaxi(10) was first reported as an upper bound by Christian Boyer in 2006 and verified as Cabtaxi(10) by Uwe Hollerbach and reported on the NMBRTHRY mailing list on May 16, 2008.