21

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(1)

2013 HMMT Guts #21: Summation of Powers of 3

Source:

Find the number of positive integers

j\leq 3^{2013}

j=\sum_{k=0}^m\left((-1)^k\cdot 3^{a_k}\right)

for some strictly increasing sequence of nonnegative integers

\{a_k\}

. For example, we may write

3=3^1

55=3^0-3^3+3^4

4

cannot be written in this form.