cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A011541 Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.

Original entry on oeis.org

2, 1729, 87539319, 6963472309248, 48988659276962496, 24153319581254312065344
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

The sequence is infinite: Fermat proved that numbers expressible as a sum of two positive integral cubes in n different ways exist for any n. Hardy and Wright give a proof in Theorem 412 of An Introduction of Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition).
A001235 gives another definition of "taxicab numbers".
David W. Wilson reports a(6) <= 8230545258248091551205888. [But see next line!]
Randall L Rathbun has shown that a(6) <= 24153319581254312065344.
C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?, 2003, show that with high probability, a(6) = 24153319581254312065344.
When negative cubes are allowed, such terms are called "Cabtaxi" numbers, cf. Boyer's web page, Wikipedia or MathWorld. - M. F. Hasler, Feb 05 2013
a(7) <= 24885189317885898975235988544. - Robert G. Wilson v, Nov 18 2012
a(8) <= 50974398750539071400590819921724352 = 58360453256^3 + 370298338396^3 = 7467391974^3 + 370779904362^3 = 39304147071^3 + 370633638081^3 = 109276817387^3 + 367589585749^3 = 208029158236^3 + 347524579016^3 = 224376246192^3 + 341075727804^3 = 234604829494^3 + 336379942682^3 = 288873662876^3 + 299512063576^3. - PoChi Su, May 16 2013
a(9) <= 136897813798023990395783317207361432493888. - PoChi Su, May 17 2013
From PoChi Su, Oct 09 2014: (Start)
The preceding bounds are not the best that are presently known.
An upper bound for a(22) was given by C. Boyer (see the C. Boyer link), namely
BTa(22)= 2^12 *3^9 * 5^9 *7^4 *11^3 *13^6 *17^3 *19^3 *31^4 *37^4 *43 *61^3 *73 *79^3 *97^3 *103^3 *109^3 *127^3 *139^3 *157 *181^3 *197^3 *397^3 *457^3 *503^3 *521^3 *607^3 *4261^3.
We also know that (97*491)^3*BTa(22) is an upper bound on a(23), corresponding to the sum x^3+y^3 with
x=2^5 *3^4 *5^3 *7 *11 *13^2 *17 *19^2 *31 *37 *61 *79 *103 *109 *127 *139 *181 *197 *397 *457 *503 *521 *607 *4261 *11836681,
y=2^4 *3^3 *5^3 *7 *11 *13^2 *17 *19 *31 *37 *61 *79 *89 *103 *109 *127 *139 *181 *197 *397 * 457 *503 * 521 *607 *4261 *81929041.
(End)
Conjecture: the number of distinct prime factors of a(n) is strictly increasing as n grows (this is not true if a(7) is equal to the upper bound given above), but never exceeds 2*n. - Sergey Pavlov, Mar 01 2017

Examples

			From _Zak Seidov_, Mar 22 2013: (Start)
Values of {b,c}, a(n) = b^3 + c^3:
n = 1: {1,1}
n = 2: {1, 12}, {9, 10}
n = 3: {167, 436}, {228, 423}, {255, 414}
n = 4: {2421, 19083}, {5436, 18948}, {10200, 18072}, {13322, 16630}
n = 5: {38787, 365757}, {107839, 362753}, {205292, 342952}, {221424, 336588}, {231518, 331954}
n = 6: {582162, 28906206}, {3064173, 28894803}, {8519281, 28657487}, {16218068, 27093208}, {17492496, 26590452}, {18289922, 26224366}. (End)

References

  • C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
  • R. K. Guy, Unsolved Problems in Number Theory, D1.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, pp. 333-334 (fifth edition), pp. 442-443 (sixth edition), see Theorem 412.
  • D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.

Links

Crossrefs

Cf. A001235, A003826, A023050, A047696, A080642 (cubefree taxicab numbers).

Formula

a(n) <= A080642(n) for n > 0, with equality for n = 1, 2 (only?). - Jonathan Sondow, Oct 25 2013
a(n) > 113*n^3 for n > 1 (a trivial bound based on the number of available cubes; 113 < (1 - 2^(-1/3))^(-3)). - Charles R Greathouse IV, Jun 18 2024

Extensions

Added a(6), confirmed by Uwe Hollerbach, communicated by Christian Schroeder, Mar 09 2008

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

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

A293650 Sum of two (possibly negative) cubes in at least 3 ways (primitive solutions).

Original entry on oeis.org

728, 3367, 4104, 5859, 46683, 65728, 68913, 101528, 124488, 134379, 152551, 155736, 165464, 168112, 184464, 195841, 205352, 289224, 333944, 342657, 402597, 439101, 622232, 625177, 684019, 754299, 757701, 842751, 845208, 1009736, 1016496, 1062936, 1073375
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Primitive means that the 6 summands are coprime. Not every term is the sum of two coprime cubes. a(1) = A047696(3).

Examples

			4104 = 18^3 - 12^3 = 16^3 + 2^3 = 15^3 + 9^3, and 18, -12, 16, 2, 15, 9 are coprime, so 4104 is in the sequence.

Links

Crossrefs

Cf. A051383 (all solutions); A003825 (positive cubes); A293651 (only coprime); A293645, A293647

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

Views

Author

Colin Mallows

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A051347, A051383.

Programs

  • PARI
    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

Formula

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

Extensions

a(6)-a(22) from Donovan Johnson, Apr 17 2010
Missing terms 42549416, 48275136, 94287375, 111209679, 124454421 added by Rosalie Fay, Oct 13 2017

A098110 Smallest number that is the difference between two positive cubes in n ways.

Original entry on oeis.org

7, 721, 3367, 4118877, 1412774811, 424910390480793
Offset: 1

Views

Author

Jeff Burch, Sep 23 2004

Keywords

Comments

a(7) <= 15490327057569000, a(8) <= 123922616460552000. - Giovanni Resta, Mar 19 2020

Examples

			Pairs (x, y) such that x^3 - y^3 = a(1), ..., a(6):
7 = (2, 1);
721 = (16, 15), (9, 2);
3367 = (34, 33), (16, 9), (15, 2)l
4118877 = (162, 51), (165, 72), (178, 115), (678, 675);
1412774811 = (1134, 357), (1155, 504), (1246, 805), (2115, 2004), (4746, 4725);
424910390480793 = (596001, 595602), (317982, 316575), (141705, 134268), (83482, 53935), (77385, 33768), (75978, 23919).

Crossrefs

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333376, A333377.
Cf. A011541,A047696.

Extensions

a(6) from Giovanni Resta, Mar 19 2020

A047697 Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) coprime cubes.

Original entry on oeis.org

1, 91, 3367, 16776487, 506433677359393
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A047696.

Programs

A125063 Numbers expressible as sum or difference of two cubes of primes in at least two ways.

Original entry on oeis.org

62540982, 105161238, 258428648, 349211772, 544861170, 772363566, 887381138, 932216922, 1154093668, 4904108118, 5498022546, 5668317018, 6058655748, 6445210590, 6507811154, 6991709634, 7088831048, 7370532162, 7426318142
Offset: 1

Views

Author

Christian Boyer (cboyer(AT)club-internet.fr), Jan 09 2007

Keywords

Comments

No known example expressible in three ways.

Examples

			62540982 = 397^3 - 31^3 = 1867^3 - 1861^3
105161238 = 193^3 + 461^3 = 709^3 - 631^3

Crossrefs

Cf. A046894, A047696.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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