OFM 2023 S2

Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes:

• if two consecutive boxes $B_i$ and $B_{i+1}$ both contain a coin, McDuck can remove the coin contained in box $B_{i+1}$ and put it on his table;
• if a box $B_i$ contains a coin, the box $B_{i+1}$ is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box $B_{i+1}$ . As a function of $k$ , which are the integers $n$ for which Scrooge McDuck can put a coin in box $B_n$ ?