Let $f(n)$ denote the sum of the prime factors of $n$ (with multiplicity).

I have been looking for pairs of consecutive numbers $n,n+1$ such that $f(n)=f(n+1)$.

Case #$1$:

- $f(8)=f(2\cdot2\cdot2)=2+2+2=6$
- $f(9)=f(3\cdot3)=3+3=6$

Case #$2$:

- $f(15)=f(3\cdot5)=3+5=8$
- $f(16)=f(2\cdot2\cdot2\cdot2)=2+2+2+2=8$

Are these the only two such pair of numbers to exist?

I think that it might be related to Catalan's conjecture.