Has work been done on looking at what happens to the exponents of the prime factorization of a number $n$ as compared to $n+1$? I am looking for published material or otherwise. For example, let $n=9=2^0\cdot{}3^2$, then,
$ 9 \;\xrightarrow{+1}\; 10 $
$ 2^0\cdot{}3^2 \;\xrightarrow{+1}\; 2^1\cdot{}3^0\cdot{}5^1 $
or looking just at the exponents,
$ [0,2,0,0,...] \;\xrightarrow{+1}\; [1,0,1,0,...] $
I realize the canonical way of reaching the latter is generating the prime factorization of $n$ and of $n+1$ separately, but has there been any research into manipulating the exponents directly instead (short-cutting around the factorization)?
For anyone who can't answer but still wants to see something interesting, check out FRACTRAN.