Hardy–Ramanujan numbers

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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, ...}

External links

• Durango Bill’s Ramanujan Numbers and The Taxicab Problem.