A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is unitary if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega(k) - 1$.
Definition. A positive integer $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}} > 1$ is hyperbolic if and only if $\sum_{i = 1}^{n} p_{i}^{-r_{i}} < 1$ or, equivalently, $\sum_{d \in D_{k}} d < k$.
See my OEIS entry.
For example, $3$, $10$ and $20$ are hyperbolic, but $30$ and $510510 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17$ are not.
Assuming my calculations are correct, I'll state the following with some confidence:
Indeed, many positive integers are hyperbolic. Of the first $10^{8}$ integers, $70334760$ are hyperbolic. Non-trivial prime powers, squares or higher, are also hyperbolic. If $k$ is a hyperbolic integer, then so are its proper, non-trivial unitary divisors; however, the same cannot be inferred for all divisors (consider the hyperbolic integer $900$ and its non-hyperbolic divisor $30$). In fact, an arbitrary product of any number of non-trivial unitary divisors of a hyperbolic integer is again hyperbolic.
The set of hyperbolic integers is closed under exponentiation, but not under addition (e.g., $10 + 20 = 30$) or under multiplication (e.g., $3 \times 10 = 30$).
Define the Prime zeta function, $\zeta_{P}(s) = \sum_{p \text{ prime}} p^{-s}$, which converges absolutely for $\mathsf{Re}(s) > 1$. Recall that the multiplicity of a prime divisor $p$ of $k$ is the largest exponent $r$ such that $p^{r}$ divides $k$ but $p^{r+1}$ does not. If the minimum multiplicity of an integer $k$ is $2$ or greater, then $k$ is hyperbolic as can be seen by the elementary bound \begin{eqnarray} \sum_{i = 1}^{n} p_{i}^{-r_{i}} < \zeta_{P}(2) \approx 0.452247 .... \end{eqnarray} Thus, the question of hyperbolicity is non-trivial only for integers with minimum multiplicity $1$.
Numerical evidence suggests that the natural density of the hyperbolic integers is greater than $0.988284 \dots$, and I conjecture that almost all integers are indeed hyperbolic (i.e., the natural density is 1).
Question: Is anything presently known about such integers? (References welcome!)
Question: Is there a simple proof showing (or refuting) that almost all integers are hyperbolic?
Thanks!