A051347 -id:A051347 - 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

A051383 Sum of two (possibly negative) cubes in at least 3 ways.

Original entry on oeis.org

728, 3367, 4104, 5824, 5859, 19656, 26936, 32832, 46592, 46683, 46872, 65728, 68913, 90909, 91000, 101528, 110808, 124488, 134379, 152551, 155736, 157248, 158193, 165464, 168112, 184464, 195841, 205352, 215488, 249704, 262656

Offset: 1

Colin Mallows

nonn

			4104 = 18^3 + (-12)^3 = 16^3 + 2^3 = 15^3 + 9^3, so 4104 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..34042

Cf. A051347, A051384.

is(n)=#thue(thueinit(z^3+1),n)>4 \\ Ralf Stephan, Oct 18 2013

A293647 Positive numbers that are the sum of two (possibly negative) cubes in at least 2 ways (primitive solutions).

Original entry on oeis.org

91, 152, 189, 217, 513, 721, 728, 999, 1027, 1729, 3087, 3367, 4104, 4706, 4921, 4977, 5256, 5859, 6832, 7657, 8587, 8911, 9919, 10621, 10712, 12663, 12691, 12824, 14911, 15093, 15561, 16120, 16263, 20683, 21014, 23058, 23877, 25669, 27937, 28063, 31519, 32984

Offset: 1

Rosalie Fay, Oct 16 2017

nonn

Primitive means that the 4 summands are coprime.
Not every term is the sum of two coprime cubes.
a(1) = A047696(2).

			189 = 4^3 + 5^3 = 6^3 + (-3)^3 and 4, 5, 6, -3 are coprime, so 189 is in the sequence.
35208 = 34^3 + (-16)^3 = 33^3 + (-9)^3 and 34, -16, 33, -9 are coprime, so 35208 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..1000 (first 352 terms from Rosalie Fay)

Cf. A051347 (all solutions); A018850 (positive cubes); A293648 (only coprime); A293645, A293650

g:= proc(s,n) local x;
  x:= s/2 + sqrt(12*n/s-3*s^2)/6;
  if not x::integer then return NULL fi;
  [x,s - x];
end proc:
filter:= proc(n)
  local pairs, i,j;
  pairs:= map(g, numtheory:-divisors(n),n);
  for i from 2 to nops(pairs) do
    for j from 1 to i-1 do
      if igcd(op(pairs[i]),op(pairs[j]))=1 then return true fi
  od od;
  false
end proc:
select(filter, [seq(seq(9*i+j,j=[1,2,7,8,9]),i=0..4000)]); # Robert Israel, Oct 22 2017

g[s_, n_] := Module[{x}, x = s/2 + Sqrt[12*n/s - 3*s^2]/6;   If[!IntegerQ[x], Return[Nothing]]; {x, s - x}];
filter[n_] := Module[{pairs, i, j}, pairs = g[#, n]& /@ Divisors[n]; For[i = 2, i <= Length[pairs], i++,For[j = 1, j <= i - 1, j++, If[GCD @@ Join[pairs[[i]], pairs[[j]]] == 1, Return[True]]]]; False];
Select[Flatten[Table[Table[9*i + j, {j, {1, 2, 7, 8, 9}}], {i, 0, 4000}]], filter] (* Jean-François Alcover, May 28 2023, after Robert Israel *)

A051384 Sum of two (possibly negative) cubes in at least 4 ways.

Original entry on oeis.org

2741256, 4118877, 6017193, 6742008, 9016488, 16776487, 21930048, 28699272, 32951016, 36875384, 42549416, 48137544, 48275136, 52324993, 53936064, 70957971, 72131904, 74013912, 87539319, 94287375, 102977784, 105651000, 111209679, 119824488, 122262264, 124454421, 134211896

Offset: 1

Colin Mallows

This sequence is infinite, since if n is in the sequence so is n*k^3 for all k > 0; thus a(n) << n^3. - Charles R Greathouse IV, Nov 29 2014

			42549416 = 348^3+74^3 = 282^3+272^3 = (-2662)^3+2664^3 = (-475)^3+531^3, so 42549416 is in the sequence. (Silverman)

Rosalie Fay, Table of n, a(n) for n = 1..100
Joseph H. Silverman, Taxicabs and sums of two cubes, Amer. Math. Monthly, 100 (1993), 331-340.

Cf. A051347, A051383.

T=thueinit('z^3+1);is(n)=my(v=thue(T, n)); #v>6 && #select(u->u[1]<=u[2],v)>3 \\ Charles R Greathouse IV, Nov 29 2014

a(1) = A047696(4). - R. J. Mathar, Aug 28 2025

a(6)-a(22) from Donovan Johnson, Apr 17 2010

Missing terms 42549416, 48275136, 94287375, 111209679, 124454421 added by Rosalie Fay, Oct 13 2017

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

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A051383 Sum of two (possibly negative) cubes in at least 3 ways.

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A293647 Positive numbers that are the sum of two (possibly negative) cubes in at least 2 ways (primitive solutions).

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A051384 Sum of two (possibly negative) cubes in at least 4 ways.

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