Chef in Evil Land

Recently, Chef has moved to an evil land. In this land, there is only one bank, which offers $N$ savings plans (numbered $1$ through $N$). Initially, Chef has $1$ coin in his account. He may choose bank plans for as long as he wishes, in any order; each plan may be chosen any number of times. For each valid $i$, whenever Chef chooses the $i$-th savings plan, the number of coins in his bank accoun