Well few days ago i asked a question on perfect numbers and Tito Piezas III answered the question in a very intriguing way which has helped me to get a lead on it.But his answer and perfect numbers made us land on a very interesting question about cubes.
The sequence A023042 on the OEIS website shows that a large percentage of $N^3$ are a sum of three positive cubes. OEIS lists only N<1770, but we can extend that:
$$\begin{array}{|c|c|} N&\text{%}\\ 2000&85.8\text{%}\\ 4000&89.8\text{%}\\ 6000&92.1\text{%}\\ 8000&93.3\text{%}\\ 10000&94.2\text{%}\\ \end{array}$$
This means that 94.2% of all N<10000 have a solution to $a^3+b^3+c^3=N^3$ in positive integers. Thus, if we pick a random N in the high end of that range, there is a very good chance that there is an a,b,c.
Now my question is that:Are there infinitely many $N^3$ (especially for prime $N$) that cannot be expressed as a sum of three positive cubes?
For example, there are no positive integers,
$$a^3+b^3+c^3=999959^3$$
even though the percentage of N<1000000 with solutions should be close to 99%.
Please help me in this question or provide me with any lead. The OEIS website link is: oeis.org/A023042
