

A011541


Taxicab, taxicab or HardyRamanujan numbers: the smallest number that is the sum of 2 positive integral cubes in n ways.


40




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. 333334 (fifth edition), pp. 442443 (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)
From Sergey Pavlov, Mar 01 2017: (Start)
Let f(n) be a(n). For 1 < n <= 6, f(n) can be written as the product of not more than x(n) distinct prime powers, where x(n) < x(n+1), 2 < x(n) <= 2n, and one of the factors is a power of 7, while, for n > 2, the second factor is 3^3. Additionally, for 1 < n < 6, f(n) can be represented as the difference between two squares (b(n))^2  (c(n))^2, where b(n) and c(n) are integer, b(n) < b(n+1), c(n) < c(n+1):
f(2)=7 *13 *19 = 55^2  36^2,
f(3)=3^3 *7 *31 *67 *223 = 9788^2  2875^2
f(4)=2^10 *3^3 *7 *13 *19 *31 *37 *127 = 2638848^2  6816^2
f(5)=2^6 *3^3 *7^4 *13 *19 *43 *73 *97 *157 = 221334064^2  329560^2
f(6)=2^6 *3^3 *7^4 *13 *19 *43 *73 *79^3 *97 *157
Conjecture: let f(n) be a(n). Then, for n > 1, f(n) can be represented as the product of not more than x(n) distinct prime powers, where x(n) <= x(n+1), 2 < x(n) <= 2n; additionally, while n > 1, f(n) can be written as the difference between two squares (b(n))^2  (c(n))^2, where b(n) and c(n) are integer, b(n) < b(n+1), c(n) < c(n+1). For n > 3, there are y "old" distinct prime powers o(1)…o(y) such that one of them is a power of 7 and the other is either a power of 3, or 3^3, and z "new" distinct prime powers n(1)…n(z) such that none of them  unlike the "old" ones  can be a divisor of a(q) while q < n.
(End)


REFERENCES

C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 2628, 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), 11961203.
R. K. Guy, Unsolved Problems in Number Theory, D1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, pp. 333334 (fifth edition), pp. 442443 (sixth edition), see Theorem 412.
PoChi Su, More Upper Bounds on Taxicab and Cabtaxi Numbers, Journal of Integer Sequences, 19 (2016), #16.4.3.
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. Boyer, New upper bounds for Taxicab and Cabtaxi numbers, JIS 11 (2008) 08.1.6
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
Ken Ono, Sarah TrebatLeder, The 1729 K3 surface, arXiv:1510.00735 [math.NT], 2015.
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), 331340.
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. A001235, A003826, A023050, A047696, A080642 (cubefree taxicab numbers).
Sequence in context: A129061 A233132 A277389 * A080642 A108331 A263076
Adjacent sequences: A011538 A011539 A011540 * A011542 A011543 A011544


KEYWORD

nonn,nice,hard,more


AUTHOR

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


EXTENSIONS

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


STATUS

approved



