Question 56
Define the sequence $a_1, a_2, a_3, \dots$ by $a_1 = 1$ and
$$
a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \dots + a_{\lfloor n/n \rfloor} + 1
$$
for $n \gt 1$. Prove that there are infinitely many $n$ such that
$$
a_n \equiv n \pmod{2^{2010}}.
$$
(This problem was suggested by Gabriel Carroll.)
$$
a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \dots + a_{\lfloor n/n \rfloor} + 1
$$
for $n \gt 1$. Prove that there are infinitely many $n$ such that
$$
a_n \equiv n \pmod{2^{2010}}.
$$
(This problem was suggested by Gabriel Carroll.)