My question is related to https://oeis.org/A269839.

It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. Famous and simplest one is $6^3 = 3^3 + 4^3 + 5^3$ which Euler noted.

I have also curiosity about solutions to $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$, although it seems that there is no any reference on it. For example, $\sum_{k=1}^{4557}k^3 = 11620^3 + 47369^3$ and there are very few known solutions to equation based on OEIS entry.

In particular, I am not sure about that there are additional probably interesting values of $n$ such as $7^{2}$, $61^{3}$ and $291^{3}$.

Question. Can be infinitely many solutions to diophantine equation $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$ ?

Any comment is welcome in order to suggest helpful ideas on question. Also any additional term is very welcome.

Thanks.