  Given an integer n>1, it is a classical Diophantine problem that whether n
can be written as a sum of two rational cubes. The study of this problem,
considering several special cases of n, has a copious history that can be
traced back to the works of Sylvester, Satg\'e, Selmer etc. and up to the
recent works of Alp\"oge-Bhargava-Shnidman. In this article, we consider the
cube sum problem for cube-free integers n which has two distinct prime factors
none of which is 3.
