士的數

第n個士的數（cabtaxi number），表示為Cabtaxi(n)，定義為能以n種方法寫成兩個或正或負或零的立方數之和的正整數中最小者。它的名字來自的士數的顛倒。對任何的n，這樣的數均存在，因為的士數對所有的n都存在。現時只有10個士的數是已知的 A047696： C a b t a x i ( 1 ) = 1 = 1 3 ± 0 3 {\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (1)&=&1&=&1^{3}\pm 0^{3}\end{matrix}}} C a b t a x i ( 2 ) = 91 = 3 3 + 4 3 = 6 3 − 5 3 {\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (2)&=&91&=&3^{3}+4^{3}\\&&&=&6^{3}-5^{3}\end{matrix}}} C a b t a x i ( 3 ) = 728 = 6 3 + 8 3 = 9 3 − 1 3 = 12 3 − 10 3 {\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (3)&=&728&=&6^{3}+8^{3}\\&&&=&9^{3}-1^{3}\\&&&=&12^{3}-10^{3}\end{matrix}}} C a b t a x i ( 4 ) = 2741256 = 108 3 + 114 3 = 140 3 − 14 3 = 168 3 − 126 3 = 207 3 − 183 3 {\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (4)&=&2741256&=&108^{3}+114^{3}\\&&&=&140^{3}-14^{3}\\&&&=&168^{3}-126^{3}\\&&&=&207^{3}-183^{3}\end{matrix}}} C a b t a x i ( 5 ) = 6017193 = 166 3 + 113 3 = 180 3 + 57 3 = 185 3 − 68 3 = 209 3 − 146 3 = 246 3 − 207 3 {\displaystyle {\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}}} C a b t a x i ( 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 {\displaystyle {\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}}} C a b t a x i ( 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 {\displaystyle {\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}}} C a b t a x i ( 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 {\displaystyle {\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}}} C a b t a x i ( 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 {\displaystyle {\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}}} C a b t a x i ( 10 ) = 933528127886302221000 = 77480130 3 − 77428260 3 = 41337660 3 − 41154750 3 = 18421650 3 − 17454840 3 = 10852660 3 − 7011550 3 = 10060050 3 − 4389840 3 = 9877140 3 − 3109470 3 = 9781317 3 − 1318317 3 = 9773330 3 − 84560 3 = 8444345 3 + 6920095 3 = 8387730 3 + 7002840 3 {\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (10)&=&933528127886302221000&=&77480130^{3}-77428260^{3}\\&&&=&41337660^{3}-41154750^{3}\\&&&=&18421650^{3}-17454840^{3}\\&&&=&10852660^{3}-7011550^{3}\\&&&=&10060050^{3}-4389840^{3}\\&&&=&9877140^{3}-3109470^{3}\\&&&=&9781317^{3}-1318317^{3}\\&&&=&9773330^{3}-84560^{3}\\&&&=&8444345^{3}+6920095^{3}\\&&&=&8387730^{3}+7002840^{3}\end{matrix}}} 在 C a b t a x i ( 3 ) {\displaystyle \mathrm {Cabtaxi} (3)} 之後，所有的士的數均用電腦來尋找。