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A011541 Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways. 29
2, 1729, 87539319, 6963472309248, 48988659276962496, 24153319581254312065344 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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".

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

REFERENCES

C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.

C. S. Calude, E. Calude and M. J. Dinneen, What is the value of Taxicab(6)?, J. Universal Computer Science, 9 (2003), 1196-1203.

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

Table of n, a(n) for n=1..6.

D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)

D. Bill, Durango Bill's Ramanujan Numbers and The Taxicab Problem

C. Boyer, New upper bounds on Taxicab and Cabtaxi numbers

C. S. & E. Calude and M. T. Dinneen, What is the value of Taxicab(6)?

U. Hollerbach, The sixth taxicab number is 24153319581254312065344, posting to the NMBRTHRY mailing list, Mar 09 2008

D. McKee, Taxicab numbers, Apr 24 2001

J. C. Meyrignac, The Taxicab Problem

I. Peterson, Math Trek, Taxicab Numbers

Randall L. Rathbun, Sixth Taxicab Number?, posting to the NMBRTHRY mailing list, Jul 16 2002

W. Schneider, Taxicab Numbers

J. Silverman, Taxicabs and Sums of Two Cubes, American Mathematical Monthly, Volume 100, Issue 4 (Apr., 1993), 331-340.

Eric Weisstein's World of Mathematics, Cubic Number

Eric Weisstein's World of Mathematics, Taxicab Number

Wikipedia, Taxicab number

D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

D. W. Wilson, Taxicab Numbers

FORMULA

a(n) <= A080642(n) for n > 0, with equality for n = 1, 2 (only?). - Jonathan Sondow, Oct 25 2013

EXAMPLE

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)

CROSSREFS

Cf. A023050, A003826, A001235, A047696, A080642 (cube-free taxicab numbers).

Sequence in context: A160224 A129061 A233132 * A080642 A108331 A162554

Adjacent sequences:  A011538 A011539 A011540 * A011542 A011543 A011544

KEYWORD

nonn,nice,hard,more

AUTHOR

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

EXTENSIONS

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.

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

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

STATUS

approved

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Last modified October 1 04:41 EDT 2014. Contains 247503 sequences.