A018787 -id:A018787 - cp's OEIS frontend

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, 373464, 402597, 439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, 684019, 704977

Offset: 1

N. J. A. Sloane

From Wikipedia: "1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 'I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."'"
A011541 gives another version of "taxicab numbers".
If n is in this sequence, then n*k^3 is also in this sequence for all k > 0. So this sequence is obviously infinite. - Altug Alkan, May 09 2016

			4104 belongs to the sequence as 4104 = 2^3 + 16^3 = 9^3 + 15^3.

R. K. Guy, Unsolved Problems in Number Theory, Section D1.
G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940, p. 12.
Ya. I. Perelman, Algebra can be fun, pp. 142-143.
H. W. Richmond, On integers which satisfy the equation t^3 +- x^3 +- y^3 +- z^3, Trans. Camb. Phil. Soc., 22 (1920), 389-403, see p. 402.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165.

Shahar Amitai, Table of n, a(n) for n = 1..30000 (terms a(1)-a(4724) from T. D. Noe, terms a(4725)-a(10000) from Zak Seidov).
Shahar Amitai, Python code to generate all taxicab numbers up to N.
J. Charles-É, Recreomath, Ramanujan's Number.
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Henk Koppelaar, Peyman Nasehpour, and Maarten Looijen, Symmetry between Series if Entangled by Sums, Preprints.org, 2024.
Istanbul Bilgi University, Ramanujan and Hardy's Taxi
Christopher Lane, The First ten Ta(2) and their double distinct cubic sums representations, Find Ramanujan's Taxi Number using JavaScript. [WayBack Machine copy]
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
J. Loy, The Hardy-Ramanujan Number.
Mia Muessig, Julia code for finding general taxicab numbers
Ken Ono and Sarah Trebat-Leder, The 1729 K3 surface, arXiv:1510.00735 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Diophantine Equation 3rd Powers
Eric Weisstein's World of Mathematics, Taxicab Number
D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

Subsequence of A003325.
Cf. A007692, A008917, A011541, A018786, A018850 (primitive solutions), A051347 (allows negatives), A343708, A360619.
Solutions in greater numbers of ways:
(>2): A018787 (A003825 for primitive, A023050 for coprime),
(>3): A023051 (A003826 for primitive),
(>4): A051167 (A155057 for primitive).

Select[Range[750000],Length[PowersRepresentations[#,2,3]]>1&] (* Harvey P. Dale, Nov 25 2014, with correction by Zak Seidov, Jul 13 2015 *)

is(n)=my(t);for(k=ceil((n/2)^(1/3)),(n-.4)^(1/3),if(ispower(n-k^3,3),if(t,return(1),t=1)));0 \\ Charles R Greathouse IV, Jul 15 2011

T=thueinit(x^3+1,1);
is(n)=my(v=thue(T,n)); sum(i=1,#v,v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ Charles R Greathouse IV, May 09 2016

A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

Original entry on oeis.org

5104, 9729, 12104, 12221, 12384, 13896, 14175, 17604, 17928, 19034, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 35216, 36288, 38259, 39339, 39376, 40041, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 41966, 42056, 42687

Offset: 1

David W. Wilson

nonn

			a(1) = A230477(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3. - _Jonathan Sondow_, Oct 24 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A008917, A018787, A025331, A025397, A025402, A025407, A230477, A343968, A344239.

Select[ Range[ 50000], 2 < Length @ Cases[ PowersRepresentations[#, 3, 3], {?Positive, ?Positive, A008917%20by%20_Jonathan%20Sondow">?Positive}] &] (* adapted from Alcover's program for A008917 by _Jonathan Sondow, Oct 24 2013 *)

is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1,k,for(b=a,k,for(c=b,k,if(a^3+b^3+c^3==n,d++))));d
n=3;while(n<50000,if(is(n)>=3,print1(n,", "));n++) \\ Derek Orr, Aug 27 2015

A008917(n) < a(n) <= A025397(n). - Jonathan Sondow, Oct 24 2013

{n: A025456(n) >= 3}. - R. J. Mathar, Jun 15 2018

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

Original entry on oeis.org

87539319, 119824488, 143604279, 175959000, 327763000, 804360375, 1840667192, 1915865217, 3080802816, 3499524728, 3623721192, 5544709352, 10458523413, 10499580728, 10833628897, 15170835645, 18778674824, 21303180171, 21989186856

Offset: 1

N. J. A. Sloane

nonn

Here n = a^3 + b^3 = c^3 + d^3 = e^3 + f^3 where gcd(a, b, c, d, e, f) = 1.

J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
R. K. Guy, Unsolved Problems in Number Theory, D1.

Ray Chandler, Table of n, a(n) for n = 1..34204
Uwe Hollerbach, Taxi, Taxi! (snapshot of lost page on web.archive.org, as of Feb. 2012).
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Diophantine Equation--3rd Powers

Cf. A018787 (all solutions).

More terms from David W. Wilson, Aug 15 1996

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

Original entry on oeis.org

6963472309248, 12625136269928, 21131226514944, 26059452841000, 55707778473984, 74213505639000, 95773976104625, 101001090159424, 159380205560856, 169049812119552, 174396242861568, 188013752349696

Offset: 1

David W. Wilson (revised Oct 15 1997)

nonn

R. K. Guy, Unsolved Problems in Number Theory, D1.

Uwe Hollerbach, Table of n, a(n) for n=1..16939
Uwe Hollerbach, Taxi, Taxi! [Original link, broken]
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

Cf. A001235, A003826, A011541, A018787, A023050, A025295, A051167, A343968, A345865.

b-file extended by Ray Chandler, Jan 19 2009

A344804 Numbers that are the sum of two cubes in exactly three ways.

Original entry on oeis.org

87539319, 119824488, 143604279, 175959000, 327763000, 700314552, 804360375, 958595904, 1148834232, 1407672000, 1840667192, 1915865217, 2363561613, 2622104000, 3080802816, 3235261176, 3499524728, 3623721192, 3877315533, 4750893000, 5544709352, 5602516416

Offset: 1

Sean A. Irvine, Jun 14 2021

nonn

			87539319 is a term because 87539319 = 167^3 + 436^3 = 22^3 + 423^3 = 255^3 + 414^3 (3 representations).
6963472309248 is not a term because 6963472309248 = 2421^3 + 19083^3 = 5436^3 + 18948^3 = 10200^3 + 18072^3 = 13322^3 + 16630^3 (4 representations).  This is the first difference between this sequence and A018787.

Sean A. Irvine, Table of n, a(n) for n = 1..1584

Cf. A018787, A025286, A025397, A343708, A345865.

cp's OEIS Frontend

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

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A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

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

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

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A344804 Numbers that are the sum of two cubes in exactly three ways.

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