Decomposition of Natural Numbers from Prime Objects

11 July 2022, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

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 $\sum_{k|n} \phi (k) = n$.

Keywords

natural numbers
prime numbers
Euler’s totient function
floor function
number theory

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