US-TST 2010 Problem 5

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and, for $n > 1$ , $$a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.$$ Prove that there are infinitely many $n$ such that $a_n \equiv n \pmod{2^{2010}}$ .