%I #25 Jul 06 2016 01:06:20
%S 349999,591408,405332018,525796270
%N Numbers n such that n*(n+1)/2 is a Taxicab number (A001235).
%C In other words, numbers n such that 0 + 1 + 2 + ... + n = a^3 + b^3 = c^3 + d^3 where (a, b) and (c, d) are distinct pairs and a, b, c, d > 0 is soluble.
%C It is known that there is no triangular number that is also a cube except 0 and 1. So if the sum of k positive cubes is a triangular number that is bigger than 1, then the minimum value of k is 2. At this point sequence focuses on that question: What are the triangular numbers that are the sum of two positive cubes in more than one way?
%C A000217(349999) = 61249825000 is the least triangular number that is also a Taxicab number.
%C a(5) > 10^9.  _Giovanni Resta_, Jul 04 2016
%e 349999 is a term because 349999*(349999+1) / 2 = 61249825000 = 820^3 + 3930^3 = 3018^3 + 3232^3.
%e 591408 is a term because 591408*(591408+1) / 2 = 174882006936 = 2070^3 + 5496^3 = 3238^3 + 5204^3.
%Y Cf. A000217, A001235.
%K nonn,more
%O 1,1
%A _Altug Alkan_, Jul 02 2016
%E a(3)a(4) from _Giovanni Resta_, Jul 04 2016
