

A011541


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


47




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!]
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(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
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)
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.
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.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.


LINKS

D. W. Wilson, Taxicab Numbers (last snapshot available on web.archive.org, as of June 2013).


FORMULA



EXAMPLE

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



KEYWORD

nonn,nice,hard,more


AUTHOR



EXTENSIONS



STATUS

approved



