sum over primes less than 'x

Question

is there a function $ f(x) $ so

$$ \sum_{p\le x}f(p)=S(x)$$

where $ S(x)=g(f(x), \pi(x) $

this means that the sum S(x) depends on the function $ f(x)$ but also on the prime number counting function

the only case is $ f(x)=0$ but can be another alternatives ?

for example, the integral of a function $ \int f(x)dx =f(x) $ means that $ f(x)=e^{x} $

but how about a sum over primes which is equal to itself for a non zero function ??

Answer 1

Henry has a good example in the comments: $$\sum_{p<x|\text{p is prime}}k = k\pi(x)$$ (where $\pi(x)$ is the prime counting function)

Answer 2

Any function $f$ that is constant on intervals $[p_i,p_{i+1})$ will do if we let $g(x,y)=\sum_{i=1}^y f(p_i)$.