The numbers 14 and 21 are quite interesting.
The prime factorisation of 14 is 2⋅7 and the prime factorisation of 14+1 is 3⋅5. Note that 3 is the prime after 2 and 5 is the prime before 7.
Similarly, the prime factorisation of 21 is 7⋅3 and the prime factorisation of 21+1 is 11⋅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≠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?
Note that exactly one of n and n+1 is even. It follows that for n to be interesting, either n=3p and n+1=2N(p) or n=2p and n+1=3P(p), where P(p) and N(p) are the previous and next primes to p respectively. Rearranging we get that p must satisfy one of the following two equations:
However, by a 1952 result of Jitsuro Nagura, for p≥25 there is always a prime between p and 65p. In particular, if p≥31 is a prime:
56p<P(p)<p<N(p)<65p But when p≥31 the following inequalities are also true:
Therefore, if p is to satisfy (1) or (2) above, it must be less than 31. This leaves a handful of cases to check for p, and we find that the only interesting numbers are 14 and 21 as conjectured.
The Nagura paper is a reference in the Wikipedia article on Bertrand's postulate. While those in the comments had saw it, sketching out the approach I use here, I already knew what to do; I did not read those comments in detail until after posting my answer.