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# Hardy-Ramanujan numbers

(Redirected from Taxi-cab numbers)

The Hardy-Ramanujan numbers (taxi-cab numbers or taxicab numbers) are the smallest positive integers that are the sum of 2 cubes of positive integers in $\scriptstyle n \,$ ways (the Hardy-Ramanujan number, i.e. the original taxi-cab number or taxicab number) being the smallest positive integer that is the sum of 2 cubes of positive integers in 2 ways).

A011541 Taxi-cab (taxicab) or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 cubes in n ways (an infinite sequence).

{2, 1729, 87539319, 6963472309248, 48988659276962496, 24153319581254312065344, ...}

## Examples

$2 = \,$
$1^3 + 1^3. \,$
$1729 = \,$
$12^3 + 1^3 = 10^3 + 9^3. \,$
$87539319 = \,$
$228^3 + 423^3 = 167^3 + 436^3 = 255^3 + 414^3. \,$
$6963472309248 = \,$
$13322^3 + 16630^3 = 10200^3 + 18072^3 = 5436^3 + 18948^3 = 2421^3 + 19083^3. \,$
$48988659276962496 = \,$
$231518^3 + 331954^3 = 221424^3 + 336588^3 = 205292^3 + 342952^3 = 107839^3 + 362753^3 = 38787^3 + 365757^3. \,$
$24153319581254312065344 = \,$
$28906206^3 + 582162^3 = 28894803^3 + 3064173^3 = 28657487^3 + 8519281^3 = \,$
$27093208^3 + 16218068^3 = 26590452^3 + 17492496^3 = 26224366^3 + 18289922^3. \,$

## Ramanujan n-way solutions

Ramanujan n-way solutions are positive integers that are the sum of 2 cubes of positive integers in n ways. (Primitive solutions share no common factors greater than 1.) One might distinguish between

• at least n-way solutions,
• exactly n-way solutions (at least n-way, but not at least (n + 1)-way solutions).

### Ramanujan 1-way solutions

A003325 Numbers that are the sum [in at least one way] of 2 positive cubes.

{2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, ...}

### Ramanujan 2-way solutions

A001235 Taxi-cab numbers: sums of 2 cubes in more than 1 way.

{1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, ...}

A018850 Numbers that are the sum of 2 cubes in more than 1 way (primitive solutions).

{1729, 4104, 20683, 39312, 40033, 64232, 65728, 134379, 149389, 171288, 195841, 216027, 327763, 402597, 439101, 443889, 515375, 684019, 704977, 805688, 842751, 920673, 955016, ...}

### Ramanujan 3-way solutions

A018787 Numbers that are the sum of two positive cubes in at least three ways (all solutions).

{87539319, 119824488, 143604279, 175959000, 327763000, 700314552, 804360375, 958595904, 1148834232, 1407672000, 1840667192, 1915865217, 2363561613, 2622104000, ...}

A003825 Numbers that are the sum of two positive cubes in at least three ways (primitive solutions).

{87539319, 119824488, 143604279, 175959000, 327763000, 804360375, 1840667192, 1915865217, 3080802816, 3499524728, 3623721192, 5544709352, 10458523413, 10499580728, ...}

### Ramanujan 4-way solutions

A023051 Numbers that are the sum of two positive cubes in at least four ways (all solutions).

{6963472309248, 12625136269928, 21131226514944, 26059452841000, 55707778473984, 74213505639000, 95773976104625, 101001090159424, 159380205560856, ...}

A003826 Numbers that are the sum of two cubes in at least four ways (primitive solutions).

{6963472309248, 12625136269928, 21131226514944, 26059452841000, 74213505639000, 95773976104625, 159380205560856, 174396242861568, 300656502205416, 376890885439488, ...}

### Ramanujan 5-way solutions

A051167 Numbers that are the sum of two positive cubes in at least five ways (all solutions).

{48988659276962496, 391909274215699968, 490593422681271000, 1322693800477987392, 3135274193725599744, 3924747381450168000, 6123582409620312000, 6355491080314102272, ...}

A155057 Numbers that are the sum of two positive cubes in at least five ways (primitive solutions).

{48988659276962496, 490593422681271000, 6355491080314102272, 27365551142421413376, 47893568195858112000, 55634997032869710456, 68243313527087529096, 265781191139199122625, ...}

### Ramanujan 6-way solutions

A?????? Numbers that are the sum of two positive cubes in at least six ways (all solutions).

{24153319581254312065344, ...}

A?????? Numbers that are the sum of two positive cubes in at least six ways (primitive solutions).

{24153319581254312065344, ...}