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taxicab numbers
Taxicab numbers are numbers that can be expressed in more than one way as a sum of two positive cubes.

For example, the smallest taxicab number is 1729, since

\[1729= 1^3 + 12^3 = 9^3 + 10^3.\]

Here the name taxicab is used in a somehow relaxed way. Other authors define the  $n-$th taxicab (or taxi-cab or Hardy-Ramanujan) number  $Ta(n)$ as the smallest number that can be written in  $n$ different way as the sum two positive integral cubes.

The name(s) of these numbers derive from a famous anecdote involving a taxicab and the famous mathematicians Hardy and Ramanujan (see the web pages linked below).

There are 870850 taxicab numbers up to  $10^{15}$. The smallest one with 3 and 4 representations are 87539319 and 6963472309248.

The first taxicab numbers are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656 more terms

Taxicab numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 760032072 48642052608 399929172552 + 6398866760832 aban 27000008 27000064 216000064 + 999960000832 abundant 4104 13832 32832 + 49933368 Achilles 28149336 225194688 760032072 + 49991146569000 alternating 216125 5832729 10589696 + 414723456 amenable 1729 4104 13832 + 998951616 apocalyptic 13832 20683 arithmetic 1729 4104 13832 + 9443761 binomial 61249825000 174882006936 c.nonagonal 61249825000 c.octagonal 49991146569 8936757492481 18476474674329 + 781111665140625 Carmichael 1729 congruent 13832 110808 134379 + 9560896 Cunningham 1729 1092728 3375001 + 729729243027001 Curzon 216125 3587409 38253878 + 185840109 cyclic 1729 20683 149389 + 9443761 d-powerful 32832 46683 525824 + 4624776 de Polignac 3242197 5772403 10765603 + 89576767 decagonal 2837477081601 deceptive 1729 deficient 1729 20683 40033 + 9443761 dig.balanced 134379 149389 195841 + 191213568 droll 34780741632 783210774528 1288175616000 + 636779367497728 Duffinian 20683 195841 327763 + 9443761 economical 110808 134379 149389 + 18673625 equidigital 110808 149389 195841 + 18673625 evil 4104 32832 39312 + 998951616 Friedman 13832 32832 216027 + 886464 frugal 134379 2991816 3628233 + 998951616 gapful 1729 32832 40033 + 99901836125 happy 110808 165464 195841 + 9443761 harmonic 46683000 Harshad 1729 4104 32832 + 9978655232 hex 195841 684019 5318677 + 980590370647771 hexagonal 61249825000 hoax 515375 1080891 1533357 + 54267003 Hogben 2436429101743 12166921836433 iban 4104 inconsummate 4104 32832 39312 + 994688 interprime 32832 402597 513000 + 98040969 junction 65728 525824 1061424 + 97867441 Lehmer 1729 39639691 788964463 + 850910580037 lucky 46683 149389 216027 + 7882245 modest 40033 320264 1080891 + 1784048168 nialpdrome 96666661000 96666661000000 nude 32832 39312 314496 + 482831496 odious 1729 13832 20683 + 996762312 palindromic 4607064 2344344434432 109514040415901 pandigital 11548089 1430867592 3928157064 + 9083647125 pernicious 1729 4104 13832 + 8872487 persistent 4315087296 5480172936 20763541981 30025986417 Poulet 1729 23226658794001 105193759783861 power 6080256576 49991146569 389136420864 + 36443545848801 powerful 28149336 225194688 760032072 + 993321157865472 practical 4104 13832 32832 + 9560896 pronic 262656 2987712 1402390152 + 688403231516160 Proth 1729 405274625 pseudoperfect 4104 13832 32832 + 994688 repunit 2436429101743 12166921836433 Ruth-Aaron 64232 4237801408 Saint-Exupery 3595213440 28761707520 97070762880 + 987335490960000 self 13832 20683 64232 + 995125824 Smith 402597 920673 1609272 + 95547816 sphenic 1729 20683 149389 + 68007673 square 6080256576 49991146569 389136420864 + 781111665140625 super Niven 513000 4104000 111321000 + 4104000000 super-d 134379 327763 515375 + 8493039 tau 110656 314496 513856 + 988048000 triangular 61249825000 174882006936 uban 27000008 1027000000 6013000000 + 98027053000000 Ulam 46683 195841 1734264 + 8587000 unprimeable 4104 171288 513856 + 9560896 untouchable 4104 39312 65728 + 994688 vampire 1456273728 1702564920 wasteful 1729 4104 13832 + 9560896 Zeisel 1729 Zuckerman 32832 8491392 11261376 + 135111375 Zumkeller 4104 13832 32832 + 65728