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A011541
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Taxicab, taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 positive cubes in n ways.
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27
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OFFSET
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1,1
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COMMENTS
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The sequence is infinite: Fermat proved that numbers expressible as a sum of two positive cubes in n different ways exist for any n, cf. Theorem 412 of Hardy & Wright, An Introduction of Theory of Numbers, p. 333-334 (fifth 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
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REFERENCES
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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.
J. Silverman, Taxicabs and Sums of Two Cubes, American Mathematical Monthly, Volume 100, Issue 4 (Apr., 1993), 331-340.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.
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LINKS
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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
Number Theory Archive, Sixth Taxicab Number?
I. Peterson, Math Trek, Taxicab Numbers
I. Peterson, Math Trek, Taxicab Numbers
Randall L. Rathbun, Posting to Number Theory List
W. Schneider, Taxicab Numbers
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
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EXAMPLE
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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)
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CROSSREFS
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Cf. A023050, A003826, A001235, A047696.
Sequence in context: A179961 A160224 A129061 * A080642 A108331 A162554
Adjacent sequences: A011538 A011539 A011540 * A011542 A011543 A011544
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KEYWORD
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nonn,nice,hard,more,changed
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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EXTENSIONS
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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 (chs@chs-kiel.de), 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
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STATUS
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approved
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