A018850 -id:A018850 - 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

A202679 Numbers that are sums of two coprime positive cubes.

Original entry on oeis.org

2, 9, 28, 35, 65, 91, 126, 133, 152, 189, 217, 341, 344, 351, 370, 407, 468, 513, 539, 559, 637, 730, 737, 793, 854, 855, 1001, 1027, 1072, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1547, 1674, 1729, 1843, 1853, 2060, 2071, 2198, 2205, 2224, 2261, 2322, 2331, 2413

Offset: 1

Arkadiusz Wesolowski, Jan 06 2012

nonn

Not a subsequence of A020898: non-cubefree members of this sequence include 152, 189, 344, 351, 513, 1072. - Robert Israel, Mar 16 2016

			28 is in the sequence since 1^3 + 3^3 = 28 and (1, 3) = 1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
R. C. Baker, Sums of two relatively prime cubes, Acta Arithmetica 129(2007), 103-146.
Kevin A. Broughan, A computational approach to characterizing the sum of two cubes, Hamilton: University of Waikato, 2001, p. 9.
P. Erdős and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), pp. 134-139. [alternate link]
P. Erdős, On the integers of the form x^k + y^k, J. London Math. Soc. 14 (1939), pp. 250-254.
Index to sequences related to sums of cubes

Subsequence of A003325.

Cf. A024670, A001235, A018850.

N:= 10000: # to get all terms <= N
S:= {2,seq(seq(x^3 + y^3, y = select(t -> igcd(t,x)=1, [$x+1 .. floor((N - x^3)^(1/3))])), x = 1 .. floor((N/2)^(1/3)))}:
sort(convert(S,list)); # Robert Israel, Mar 15 2016

nn = 2500; Union[Flatten[Table[If[CoprimeQ[x, y] == True, x^3 + y^3, {}], {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]]
Select[Range@ 2500, Length[PowersRepresentations[#, 2, 3] /. {{0, } -> Nothing, {a, b_} /; ! CoprimeQ[a, b] -> Nothing}] > 0 &] (* Michael De Vlieger, Mar 15 2016 *)

is(n)=for(k=1,(n\2+.5)^(1/3),if(gcd(k,n)==1&&ispower(n-k^3, 3), return(1)));0 \\ Charles R Greathouse IV, Apr 13 2012

list(lim)=my(v=List()); forstep(x=1, lim^(1/3), 2, forstep(y=2,(lim-x^3+.5)^(1/3), 2, if(gcd(x,y)==1, listput(v,x^3+y^3))); forstep(y=1, min((lim-x^3+.5)^(1/3),x), 2, if(gcd(x,y)==1, listput(v,x^3+y^3)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Mahler shows that a(n) < kn^(3/2) for some k. Erdős later gives an elementary proof. - Charles R Greathouse IV, Dec 05 2012

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 *)

A384106 Numbers representable as the sum of 2 cubes in at least 2 ways generated by a parameterized formula involving (7+4sqrt(3))^n and (7-4sqrt(3))^n.

Original entry on oeis.org

1009736, 2714690888, 7334904115448, 19818905563705976, 53550675461437475048, 144693905277386048024168, 390962878508814502873889816, 1056203940519850679825934312168, 2853755704387709706549646191448888, 7710144396612746633517746345789261976

Offset: 1

Jamal Agbanwa, May 19 2025

nonn

A rapidly growing sequence of integers, each equal to x(n)^3 + y(n)^3 = u(n)^3 + w(n)^3 for distinct positive integers x(n), y(n), u(n), w(n), generated from a parameterized expression. Values omit small classical examples (like 1729) and begin at much larger values and is therefore a parameterized subset of solutions to A001235.

			For n = 7, a(7) = x(n)^3 + y(n)^3 = ((-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^7 + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^7)/4 + 3)^3 + ((-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^7 + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^7)/4)^3 = 390962878508814502873889816.

Jamal Agbanwa, A Closed-Form Symbolic Generator: A^n + B^n = C^n + D^n, n = 2, 3, Preprint, 2025. See also arXiv:2506.19173 [math.GM], 2025, p. 8.

Cf. A011541, A018850.

Subset of A001235.

a(n) = x(n)^3 + y(n)^3 = u(n)^3 + w(n)^3 where:
  x(n) = (-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n)/4 + 3,
  y(n) = (-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n)/4,
  u(n) = (-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n)/4, abd
  w(n) = (-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n)/4 + 9.

cp's OEIS Frontend

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

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A202679 Numbers that are sums of two coprime positive cubes.

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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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A384106 Numbers representable as the sum of 2 cubes in at least 2 ways generated by a parameterized formula involving (7+4*sqrt(3))^n and (7-4*sqrt(3))^n.

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A384106 Numbers representable as the sum of 2 cubes in at least 2 ways generated by a parameterized formula involving (7+4sqrt(3))^n and (7-4sqrt(3))^n.