This looks like an easy question but I can't mathematically generalize it. I need to form a pattern and prove it. So far I have observed the pattern of [153, 165033, 166500333, 166650003333, ...], but I can't seem to mathematically come to a generalization
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1Flagged for moderator attention. Like the other one you posted, this question is part of the PROMYS 2017 application problem set which is still ongoing with a submission deadline of April 1st 2017. At the risk of repeating my other comment: this site doesn't condone cheating. – dxiv Mar 08 '17 at 05:37
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(Using the ``floor'' function) I think this is an expression for your pattern for a given positive integer $n$
$$\left\lfloor \frac{10^n}{6}\right\rfloor^3+\left\lfloor\frac{10^n}{2}\right\rfloor^3+\left\lfloor\frac{10^n}{3}\right\rfloor^3= \left\lfloor \frac{10^n}{6}\right\rfloor 10^{2n} + \left\lfloor \frac{10^n}{2}\right\rfloor 10^{n}+\left\lfloor \frac{10^n}{3}\right\rfloor.$$
Maple verified that this works for $n=1, 2, \ldots, 10$. I didn't try to prove it.
