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From Wikipedia: "1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan. In Hardy's words: 'I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."'"

A011541 gives another version of "taxicab numbers".

If n is in this sequence, then n*k^3 is also in this sequence for all k > 0. So this sequence is obviously infinite. - Altug Alkan, May 09 2016

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)

Although this sequence has keyword "bref", this sequence is infinite since if n is in this sequence, then n*k^3 is in this sequence for all k > 0. - Altug Alkan, May 10 2016

Primitive means that the 6 summands are coprime. Not every term is the sum of two coprime cubes. a(1) = A047696(3).