Cabtaxi number

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In mathematics, 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 (since taxicab numbers exist for all n); however, only 9 are known (sequence A047696 in OEIS):

\begin{matrix}\mathrm{Cabtaxi}(1)&=&1&=&1^3 \pm 0^3\end{matrix}
\begin{matrix}\mathrm{Cabtaxi}(2)&=&91&=&3^3 + 4^3 \\&&&=&6^3 - 5^3\end{matrix}
\begin{matrix}\mathrm{Cabtaxi}(3)&=&728&=&6^3 + 8^3 \\&&&=&9^3 - 1^3 \\&&&=&12^3 - 10^3\end{matrix}
\begin{matrix}\mathrm{Cabtaxi}(4)&=&2741256&=&108^3 + 114^3 \\&&&=&140^3 - 14^3 \\&&&=&168^3 - 126^3 \\&&&=&207^3 - 183^3\end{matrix}
\begin{matrix}\mathrm{Cabtaxi}(5)&=&6017193&=&166^3 + 113^3 \\&&&=&180^3 + 57^3 \\&&&=&185^3 - 68^3 \\&&&=&209^3 - 146^3 \\&&&=&246^3 - 207^3\end{matrix}
\begin{matrix}\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\end{matrix}
\begin{matrix}\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\end{matrix}
\begin{matrix}\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\end{matrix}
\begin{matrix}\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\end{matrix}

Cabtaxi(5), Cabtaxi(6) and Cabtaxi(7) were found by Randall L. Rathbun; Cabtaxi(8) was found by Daniel J. Bernstein; Cabtaxi(9) was found by Duncan Moore, using Bernstein's method.

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