I am looking for a function that for any natural number n returns a natural number m. Inverse function should exist for any $m=F(n), n \in \mathbb{N}$. Sequence should be simple enough for a person to calculate first 1000 members on paper in a day. Sequence should have little apparent order.
Examples not matching criteria
m=0*n+14 No inverse function
m=34*n*n-42: Easy to guess this law
m= mod(a^n, d) : modular arithmetic is periodic and inverse function returns values (1,n)
https://oeis.org/A000055 : Close, but too complex to compute
Prime numbers: Best example so far. Pros
- Not trivial
- Defined for all n
- Easily calculatable
Cons
- Well known
Which sequence behaves as nice as prime counting, but is relatively less studied?
Update: some cellular automata sequences seem promissing, but they
- Grow fast (nearly exponential growth)
- Hard to remember (error prone procedure)