For a positive integer , define $$f(n):=\omega(F_n+n)$$ where $F_n$ denotes the $n$-th fibonacci-number and $\omega(n)$ denotes the number of distinct prime factors of $n$.
Define $g(k)$ as the smallest positive integer $g$ with $f(g)=k$ , if such an integer exists and undefined otherwise.
Is g a total function ? In other words , can $F_n+n$ have any given positive number of disctint prime factors ?
The values of $g$ upto $k=10$ :
gp > for(k=1,10,s=1;while(omega(fibonacci(s)+s)<>k,s=s+1);print(k," ",s))
1 1
2 5
3 12
4 19
5 36
6 47
7 89
8 203
9 190
10 144
gp >
Further , we have $g(11)=358$ and $g(k)>358$ for every $k>11$ since $n=358$ is the first case with at least $11$ prime factors. Similar , we have $g(12)=420$ and $g(k)>420$ for $k>12$