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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

• 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.
• 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.

Crossrefs

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).

Programs

Mathematica

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

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