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
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)
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)
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)
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