cp's OEIS Frontend

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

%I A047696 #89 Mar 20 2025 10:36:39
%S A047696 1,91,728,2741256,6017193,1412774811,11302198488,137513849003496,
%T A047696 424910390480793000,933528127886302221000
%N A047696 Smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes.
%C A047696 Sometimes called cab-taxi (or cabtaxi) numbers.
%C A047696 For a(10), see the C. Boyer link.
%C A047696 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
%C A047696 From _PoChi Su_, Aug 14 2014: (Start)
%C A047696 An upper bound of a(42) was given by C. Boyer (see the C. Boyer link), denoted by
%C A047696 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*
%C A047696         61^3*67^3*73*79^3*97^3*101^3*109^3*139^3*157*163^3*181^3*
%C A047696         193^3*223^3*229^3*307^3*397^3*457^3.
%C A047696 We show that 503^3*BCa(42) is an upper bound of a(43) with an additional sum of x^3+y^3, with
%C A047696 x=2^4*3^3*5^5*7*11*13^2*17*29*37*43*61*67*79*97*101*109*139*163*
%C A047696   181*193*223*229*307*397*457*2110099,
%C A047696 y=2^3*3^4*5^3*7*11*13^2*17*29*37*41*43*61*67*79*97*101*109*139*163*
%C A047696   181*193*223*229*307*397*457*176899.
%C A047696 (End)
%C A047696 From _PoChi Su_, Aug 29 2014: (Start)
%C A047696 An upper bound of a(43) was given by _PoChi Su_, denoted by
%C A047696 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*
%C A047696         61^3*67^3*73*79^3*97^3*101^3*109^3*139^3*157*163^3*181^3*
%C A047696         193^3*223^3*229^3*307^3*397^3*457^3*503^3.
%C A047696 We show that 1307^3*SCa(43) is an upper bound of a(44) with an additional sum of x^3+y^3, with
%C A047696 x=2^3*3^4*5^3*7^2*11*13^2*17*19*23*29*37*43*61*79*101*109*139*163*
%C A047696   181*193*223*229*307*353*397*457*503*826583,
%C A047696 y=-2^7*3^3*5^3*7^2*11*13^2*17*19^2*29*37*43*61*79*101*109*139*163*
%C A047696   181*193*223*229*307*397*457*503*58882897.
%C A047696 (End)
%C A047696 From _Sergey Pavlov_, Feb 18 2017: (Start)
%C A047696 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):
%C A047696 a(2) = 7 * 13 = 10^2 - 3^2 = 91,
%C A047696 a(3) = 2^3 * 7 * 13 = 33^2 - 19^2,
%C A047696 a(4) = 2^3 * 3^3 * 7^3 * 37 = 1659^2 - 105^2,
%C A047696 a(5) = 3^3 * 7 * 13 * 31 * 79 = 2477^2 - 344^2,
%C A047696 a(6) = 3^3 * 7^4 * 19 * 31 * 37 = 37590^2 - 483^2,
%C A047696 a(7) = 2^3 * 3^3 * 7^4 * 19 * 31 * 37 = 106477^2 - 5929^2,
%C A047696 a(8) = 2^3 * 3^3 * 7^4 * 19 * 23^3 * 31 * 37 = 11736739^2 - 487025^2,
%C A047696 a(9) = 2^3 * 3^3 * 5^3 * 7^4 * 19 * 31 * 37 * 67^3 = 651858879^2 - 3099621^2,
%C A047696 a(10) = 2^3 * 3^3 * 5^3 * 7^4 * 13^3 * 19 * 31 * 37 * 67^3.
%C A047696 (End)
%D A047696 C. Boyer, "Les nombres Taxicabs", in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
%D A047696 R. K. Guy, Unsolved Problems in Number Theory, Section D1.
%H A047696 Daniel J. Bernstein, <a href="http://cr.yp.to/papers.html#sortedsums">Enumerating solutions to p(a) + q(b) = r(c) + s(d)</a>
%H A047696 Daniel J. Bernstein, <a href="http://pobox.com/~djb/papers/sortedsums.dvi">Enumerating solutions to p(a) + q(b) = r(c) + s(d)</a>
%H A047696 Christian Boyer, <a href="http://www.christianboyer.com/taxicab">New upper bounds on Taxicab and Cabtaxi numbers</a>.
%H A047696 Christian Boyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Boyer/boyer.html">New upper bounds for Taxicab and Cabtaxi numbers</a>, JIS 11 (2008) 08.1.6.
%H A047696 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_22">On Some Special Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
%H A047696 Uwe Hollerbach, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;19ef9d82.0805">The tenth cabtaxi number is 933528127886302221000</a>, May 14, 2008.
%H A047696 Uwe Hollerbach, <a href="http://www.korgwal.com/ramanujan/">Taxi, Taxi!</a> [Original link, broken]
%H A047696 Uwe Hollerbach, <a href="http://web.archive.org/web/20120203221114/http://www.korgwal.com/ramanujan">Taxi, Taxi!</a> [Replacement link to Wayback Machine]
%H A047696 Uwe Hollerbach, <a href="/A003825/a003825.html">Taxi! Taxi!</a> [Cached copy from Wayback Machine, html version of top page only]
%H A047696 Po-Chi Su, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Su/su3.html">More Upper Bounds on Taxicab and Cabtaxi Numbers</a>, Journal of Integer Sequences, 19 (2016), #16.4.3.
%H A047696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TaxicabNumber.html">Taxicab Numbers</a>
%H A047696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CabtaxiNumber.html">Cabtaxi Number</a>
%H A047696 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cabtaxi_number">Cabtaxi number</a>
%e A047696 91 = 6^3 - 5^3 = 4^3 + 3^3 (in two ways).
%e A047696 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.
%Y A047696 Cf. A011541, A047697.
%K A047696 nonn,nice,more,hard
%O A047696 1,2
%A A047696 _N. J. A. Sloane_
%E A047696 a(9) (which was found on Jan 31 2005) from Duncan Moore (Duncan.Moore(AT)nnc.co.uk), Feb 01 2005