For prime $p$ we can observe the $p$ remainders $b_1,...,b_p$ and $p$ quotients $a_1,...,a_p$ by writing $p$ as $p=a_k \cdot k+b_k$ for $1 \leq k \leq p$.
Because of the familiar division-with-remainder-theorem we have $0 \leq b_k <p$
But since $p=(\frac{p-1}{2}-w)+(\frac {p+1}{2}+w) \cdot 1$ for $w=0,...,\frac{p-1}{2}$ the pattern, if it is to be observed, should(?) include only the sum $$r(p)=\sum_{k=1}^{\frac{p-1}{2}}b_k$$
Also the sum of quotients could be studied, that is, the sum $$q(p)=\sum_{k=1}^{\frac{p-1}{2}}a_k$$
Do you think that really there are some not-so-hard-to-observe patterns (hopefully, some which would characterize primes) of $r(p)$ and $q(p)$ and did you observe some of them?
If you decide to do some computations, inform me whether you found some interesting results.
Also, do you know how to prove something non-trivial about $r(p)$ and $q(p)$?
Your thoughts about these two simple sequences?