Using $n=a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$ we see that $a+b\ge19$ If $a+b=19$ then to minimise $a^3+b^3$ their difference should be as small as possible, which is easy to check by looking at the function $x^3+\left(19-x\right)^3$. First checking $\left(a,\ b\right)=\left(9,\ 10\right)$ this doesn't work as $9^3+10^3=7\cdot13\cdot19$ Then checking the next smallest $\left(a,\ b\right)=\left(8,\ 11\right)$ we see that $8^3+11^3=19\cdot97$ which does work and thus $n=8^3+11^3=1843$ is the smallest possible $n$ Obviously if $a+b>19$ then it is obvious that the smallest $n$ would be larger than $8^3 +11^3$