Find the formula for even integers that can be expressed as $p_{1}p_{2}+p_{3}p_{4}$, where $p_{i}$ denote prime numbers. Then, solve the generalized task: find even integers “S-i” that can be expressed as $p_{1}p_{2}...p_{i}+p_{i+1}...p_{2i}$.

This task came from the need to generalize the Goldbach Conjecture. I claimed that every even number can be shown as sum of numbers of the same dimension, where dimension of $a$ is $k$ iff $u(a)=k$. This I proved incorrect, but consider this path fruitful.
The equation of Goldbach conjecture looks ugly, hence I consider there must be decent generalization of that. Lets reverse it, and verify the converse.

For every two prime numbers (lets forget about 2), there sum is even.

2n = p1 + p2=>n = (p1 + p2)/2

Lets assume p1=p2, then 2n=2p1=>n=p1. This is straightforward that only n such that u(n)=2 allow this condition (p1=p2). Lets now assume that we analyze the case where u(n)>2. Then we see that p1!=p2. One sees that p1<n and p2<n is not possible.

So, there are is possible option: p1<n and p2>n (with assumption that p2>p1 without loss of generality). Lets now assume that every interger n is characterized by r = min(n-p1)= min(p2-n). I should take a closer look at “r” number as well as create characteristic equations for every n to find all the possible r’s for the integer number.