On $sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y ge 1$

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. Largest solution that I found as below. $$sum_{k=1}^{524^{3}}k^3 = 107131073934081017703266616960000 = 32201140654^3 + 41934379346^3$$

In particular, I am not sure about that there are additional probably interesting values of $$n$$ such as $$7^{2}$$, $$61^{3}$$, $$293^{3}$$, $$440^{3}$$ and $$524^{3}$$.

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.

MathOverflow Asked by Alkan on January 26, 2021

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