Decomposition of Natural Numbers from Prime Objects

We decompose natural numbers from structure which prime numbers have, as its starting point. With the decomposition, we can find a general law by categorization, which is in a power set and also in structure which prime numbers have, and we know that it limits the framework of structure about product and sum of natural numbers. In other words, $\sum_{k=1}^{n} \phi (k) \times [\frac{n}{k}] = \frac{n(n+1)}{2}$ holds, and it is equivalent to a basic formula of sum of divisors $\sum_{k|n} \phi (k) = n$.