3 added 175 characters in body

How about this one: $f(n)=1$ if $2^{2^n}+1$ is prime, and $f(n)=0$ otherwise.

Or: $f(n)$ is the largest prime factor of $2^{2^n}+1$.

Added 1: Inspired by Barry Cipra's comment, here is a function which is unknown in principle, not just in practice: Let $r(m)$ denote the number of lattice points on the sphere $x^2+y^2+z^2=4m+1$, and define $f(n)$ as the largest $m$ with $r(m) < n\cdot m^{1/3}$.

Added 2: Here is another function, inspired by Barry Cipra's example: $f(n)$ is the $n$-th positive integer which is a sum of three cubes (including negative cubes).

2 added 270 characters in body

How about this one: $f(n)=1$ if $2^{2^n}+1$ is prime, and $f(n)=0$ otherwise.

Or: $f(n)$ is the largest prime factor of $2^{2^n}+1$.

Added: Inspired by Barry Cipra's comment, here is a function which is unknown in principle, not just in practice: Let $r(m)$ denote the number of lattice points on the sphere $x^2+y^2+z^2=4m+1$, and define $f(n)$ as the largest $m$ with $r(m) < n\cdot m^{1/3}$.

1

How about this one: $f(n)=1$ if $2^{2^n}+1$ is prime, and $f(n)=0$ otherwise.

Or: $f(n)$ is the largest prime factor of $2^{2^n}+1$.