A047696 Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes.

1, 91, 728, 2741256, 6017193, 1412774811, 11302198488, 137513849003496, 424910390480793000, 933528127886302221000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• more
• hard

Comments

Sometimes called cab-taxi (or cabtaxi) numbers.

For a(10), see the C. Boyer link.

Christian Boyer: After his recent work on Taxicab(6) confirming the number found as an upper bound by Randall Rathbun in 2002, Uwe Hollerbach (USA) confirmed this week that my upper bound constructed in Dec 2006 is really Cabtaxi(10). See his announcement. - Jonathan Vos Post, Jul 08 2008

From PoChi Su, Aug 14 2014: (Start)

An upper bound of a(42) was given by C. Boyer (see the C. Boyer link), denoted by

BCa(42)= 2^9*3^9*5^9*7^7*11^3*13^6*17^3*19^3*29^3*31*37^4*43^4*

61^3*67^3*73*79^3*97^3*101^3*109^3*139^3*157*163^3*181^3*

193^3*223^3*229^3*307^3*397^3*457^3.

We show that 503^3*BCa(42) is an upper bound of a(43) with an additional sum of x^3+y^3, with

x=2^4*3^3*5^5*7*11*13^2*17*29*37*43*61*67*79*97*101*109*139*163*

181*193*223*229*307*397*457*2110099,

y=2^3*3^4*5^3*7*11*13^2*17*29*37*41*43*61*67*79*97*101*109*139*163*

181*193*223*229*307*397*457*176899.

(End)

From PoChi Su, Aug 29 2014: (Start)

An upper bound of a(43) was given by PoChi Su, denoted by

SCa(43)= 2^9*3^9*5^9*7^7*11^3*13^6*17^3*19^3*29^3*31*37^4*43^4*

61^3*67^3*73*79^3*97^3*101^3*109^3*139^3*157*163^3*181^3*

193^3*223^3*229^3*307^3*397^3*457^3*503^3.

We show that 1307^3*SCa(43) is an upper bound of a(44) with an additional sum of x^3+y^3, with

x=2^3*3^4*5^3*7^2*11*13^2*17*19*23*29*37*43*61*79*101*109*139*163*

181*193*223*229*307*353*397*457*503*826583,

y=-2^7*3^3*5^3*7^2*11*13^2*17*19^2*29*37*43*61*79*101*109*139*163*

181*193*223*229*307*397*457*503*58882897.

(End)

From Sergey Pavlov, Feb 18 2017: (Start)

For 1 < n <= 10, each a(n) can be written as the product of not more than n distinct prime powers where one of the factors is a power of 7. For 1 < n <= 9, a(n) can be represented as the difference between two squares, b(n)^2 - c(n)^2, where b(n), c(n) are integers, b(n+1) > b(n), and c(n+1) > c(n):

a(2) = 7 * 13 = 10^2 - 3^2 = 91,

a(3) = 2^3 * 7 * 13 = 33^2 - 19^2,

a(4) = 2^3 * 3^3 * 7^3 * 37 = 1659^2 - 105^2,

a(5) = 3^3 * 7 * 13 * 31 * 79 = 2477^2 - 344^2,

a(6) = 3^3 * 7^4 * 19 * 31 * 37 = 37590^2 - 483^2,

a(7) = 2^3 * 3^3 * 7^4 * 19 * 31 * 37 = 106477^2 - 5929^2,

a(8) = 2^3 * 3^3 * 7^4 * 19 * 23^3 * 31 * 37 = 11736739^2 - 487025^2,

a(9) = 2^3 * 3^3 * 5^3 * 7^4 * 19 * 31 * 37 * 67^3 = 651858879^2 - 3099621^2,

a(10) = 2^3 * 3^3 * 5^3 * 7^4 * 13^3 * 19 * 31 * 37 * 67^3.

(End)

Examples

			91 = 6^3 - 5^3 = 4^3 + 3^3 (in two ways).
Cabtaxi(9)=424910390480793000 = 645210^3 + 538680^3 = 649565^3 + 532315^3 = 752409^3 - 101409^3 = 759780^3 - 239190^3 = 773850^3 - 337680^3 = 834820^3 - 539350^3 = 1417050^3 - 1342680^3 = 3179820^3 - 3165750^3 = 5960010^3 - 5956020^3.
		

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, Section D1.

Links

• Daniel J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
• Daniel J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
• Christian Boyer, New upper bounds on Taxicab and Cabtaxi numbers.
• Christian Boyer, New upper bounds for Taxicab and Cabtaxi numbers, JIS 11 (2008) 08.1.6.
• Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
• Uwe Hollerbach, The tenth cabtaxi number is 933528127886302221000, May 14, 2008.
• Uwe Hollerbach, Taxi, Taxi! [Original link, broken]
• Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
• Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
• Po-Chi Su, More Upper Bounds on Taxicab and Cabtaxi Numbers, Journal of Integer Sequences, 19 (2016), #16.4.3.
• Eric Weisstein's World of Mathematics, Taxicab Numbers
• Eric Weisstein's World of Mathematics, Cabtaxi Number
• Wikipedia, Cabtaxi number

Crossrefs

Cf. A011541, A047697.

Extensions

a(9) (which was found on Jan 31 2005) from Duncan Moore (Duncan.Moore(AT)nnc.co.uk), Feb 01 2005