Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us call it prime giving function
It can be shown that $d(x)$ -number of divisors of $x$ is one such prime giving function by the combination of Bernard's postulate and prime no. theorem . The question is which are the other such prime giving functions ? And what are the proofs that they posses such properties?