A001235

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

DATA

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

PYTHON IMPLEMENTATION

No Python implementation recorded in the local database yet.

PROGRAMS

Mathematica

Select[Range[750000],Length[PowersRepresentations[#,2,3]]>1&] (* _Harvey P. Dale_, Nov 25 2014, with correction by _Zak Seidov_, Jul 13 2015 *)

Other Programs

• (PARI) :

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

• (PARI) :

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

COMMENTS

• 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

EXAMPLES

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

REFERENCES

• Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 307.
• 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.

LINKS

• 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.
• Istanbul Bilgi University, Ramanujan and Hardy's Taxi
• Henk Koppelaar, Peyman Nasehpour, and Maarten Looijen, Symmetry between Series if Entangled by Sums, Preprints.org, 2024.
• 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.

CROSSREFERENCES

• 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).
• See also the official entry at https://oeis.org/A001235

METADATA

• OFFSET : 1,1
• KEYWORDS : nonn,nice,changed
• AUTHOR : N. J. A. Sloane
• IDENTIFICATION : #149 Oct 19 2026 15:58:05