1. $[\mathbf{3}]$ What is the smallest positive integer $n$ which cannot be written in any of the following forms?
- $n=1+2+\cdots+k$ for a positive integer $k$.
- $n=p^{k}$ for a prime number $p$ and integer $k$.
- $n=p+1$ for a prime number $p$.
Answer: $\quad 22$ Consider $1,2,3,4,5,7,8,9,11,13,16,17,19$ are in the form $p^{k}$. So we are left with $6,10,12,14,15,18,20,21,22, \ldots$
Next, $6,12,14,18,20$ are in the form $p+1$, so we are left with $10,15,21,22, \ldots$
Finally, $10,15,21$ are in the form $n=1+2+\cdots+k$, so we are left with $22, \ldots$
Since $22=2 \cdot 11$ is not a prime power, $22-1=21$ is not prime, and $1+2+\cdots+6=21<22<28=$ $1+2+\cdots+7,22$ is the smallest number not in the three forms, as desired.