$$\frac{\sum_{i=\pi (n )+1}^{\pi(n+1 )}\sum _{j=1}^{{\bigl\lfloor\frac {\ln (n +1) }{\ln ( p_{{i}} ) }}\bigr\rfloor +1}\Bigl\lfloor {\frac {n+1}{{p_{{i}}}^{j}}} \Bigr\rfloor \ln
( p_{{i}})}{\ln(n+1)}=\delta(f(n),g(n))$$
$${\{f(n),g(n)}\} \subset \mathbb N$$
Where $\delta$ is the Kronecker delta function.