Using natural numbers $1,2,...n,$ in how many ways can the number $n$ be formed from the sum of **one** or more smaller natural numbers?

Question

Using natural numbers $1,2,...n,$ in how many ways can the number $n$ be formed from the sum of one or more smaller natural numbers?

I thought it would be an easy problem but i couldn't figure it out. Example: For n = 4, we have 4$,1 + 3, 2 + 2, 1 + 1 + 2,$1 + 1 + 1 + 1 for a total of $5.$

Answer 1

The answer is complicated and you've missed $4=1+1+1+1$ in your question. You are looking for partitions - the link has a good bibliography. The exact formula is surprisingly complicated. Your question involves subtracting 1 from the number of partitions as conventionally calculated.