For $\,p_n\gt2\,$ let's define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime.
The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 - sum of the remainders when the $\,n$-th prime is divided by primes up to the $\,(n-1)$-th prime) are:
$S(3)=1\;\;\;(p_2=3)$
$S(5)=3\;\;\;(p_3=5)$
$S(7)=4\;\;\;(p_4=7)$
$S(11)=8\;\;\;(p_5=11)$
$S(13)=13\;\;\;(p_6=13)$
$S(17)=18\;\;\;(p_7=17)$
$S(19)=27\;\;\;(p_8=19)$
$S(23)=29\;\;\;(p_9=23)$
$S(29)=46\;\;\;(p_{10}=29)$
The graphical representation of $\,S(p_n)\,$ up to $\,n=1229\,$ is the following:
The growing of $\,S(p_n)\,$ is well estimated by $$S(p_n)\sim n^2\cdot\log\log(n)\;\;\;\;\;\;\;\;(1)$$
Further, the modular equation $$S(p_n)\equiv0\mod p_n\;\;\;\;\;\;\;\;(2)$$ has, up to $\,n=78498$, only the following solutions:
$p_6=13\;\;\;S(13)=13$
$p_{39}=167\;\;\;S(167)=1002$
$p_{333}=2239\;\;\;S(2239)=123145$
$p_{36931}=439867\;\;\;S(439867)=2789196647$
I ask if the previous results, (1) and (2), can be motivated from a theoretical point of view.
Many thanks.