The numbers 14 and 21 are quite interesting.
The prime factorisation of 14 is $2\cdot 7$ and the prime factorisation of $14+1$ is $3\cdot 5$. Note that 3 is the prime after 2 and 5 is the prime before 7.
Similarly, the prime factorisation of 21 is $7\cdot 3$ and the prime factorisation of $21+1$ is $11\cdot 2$. Again, 11 is the prime after 7 and 2 is the prime before 3.
In other words, they both satisfy the following definition:
Definition: A positive integer $n$ is called interesting if it has a prime factorisation $n=pq$ with $p\ne q$ such that the prime factorisation of $n+1$ is $p'q'$ where $p'$ is the prime after $p$ and $q'$ the prime before $q$.
Are there other interesting numbers?