When working on a problem in probabilistic number theory, I was trying to prove the existence of a coupling of some number-theoretical variables subject to some constraints. In the process (and before I learned to apply Hall's/Strassen's Theorem), I created some function $P$ which measures compositeness. In particular, $P:\mathbb{N}\to [0,1)$, $$P\left(j\right) = 1 - \frac{1+\sum_{p \vert j}\left(p-1\right)}{j},$$ where the sum is over the distinct prime factors of $j$.
Some properties of $P$ are
- $P\left(j\right)=0$ if and only if $j$ is 1 or $j$ is prime;
- Of all composite numbers, $P$ says that $6$ is the least composite in the sense that $P\left(6\right)=1/3$ is the smallest nonzero value for $P$. In particular, $P$ claims that $2\cdot 3$ is less composite than $2\cdot2$ since $P\left(4\right)=1/2$.
- As expected, there is no largest $P$ value (i.e., no "most composite number"); e.g., $P\left(2^n\right) = 1-1/2^{n-1} \to 1$.
- In addition, I have a method that naturally extends $P$ to a multivariate function that can be used to measure relative compositeness among lists (e.g., $P\left(4, 8\right) = 1/2 > 1/3 = P\left(4, 9\right)$. However, the formula becomes tedious even in the 2-dimensional case: if $a=\prod p_{k}^{a_k}$ and $b=\prod p_{k}^{b_k}$, then
$$P(a,b) = 1 - \left(\frac{1}{a}\left(\sum_{a_{k} \ge 1}\left(p_{k}- 1\right)\right)+1\right) - \left(\frac{1}{b}\left(\sum_{b_{k} \ge 1}\left(p_{k}- 1\right)\right)+1\right)\\ \phantom{P(a,b) = } + \frac{1}{\text{lcm}(a,b)}\left(\sum_{a_k - b_k \ge 1}\left(p_k - 1\right)+1\right)\left(\sum_{b_k - a_k \ge 1}\left(p_k - 1\right)+1\right)$$
- Poset Structure: Say $a \preceq b$ if $\frac{a}{\gcd(a,i)}$ is composite implies $\frac{b}{\gcd(b,i)}$ is composite. If $a \preceq b$, then $P(a) \le P(b)$ and $P(a,b) = P(a)$.
- $P$ is not a measure since it is not additive: $P(4,9) = 1/3 < 1/2 + 2/3 = P(4) + P(9)$
Is there an avenue in number theory where such a function or its properties would be of interest? Admittedly, I came about this function in a round-about way.
