This is a conjecture inspired by the relative but piecewise smoothness of the sequence https://oeis.org/A068901/graph?png=1, about the least number $a_n$, such that $n\mid p_n+a_n$, where $p_n$ is the $n$-th prime.
Conjecture:
$n\mid p_n+a \wedge 0≤a<n \Rightarrow \exists b\in\mathbb Z: n+1\mid p_{n+1}+b$, where $|a-b|<2\cdot\sqrt n$.
Are there counter-examples, or can the conjecture be proved? Tested for all primes $p_n<100,000,000$.
The conjecture badly describe the real smothness. Making some more experiments lead me to comment my question with an other conjecture:
As above, given $\varepsilon>0$ there exist $N$ such that $n>N \Rightarrow |a-b|<\varepsilon\sqrt n$.