Let $X$ denote the largest subset of odd integers with the property that
every exponent in the prime factorization of any $x \in X$ belongs to $X$.


The conjecture states that the density of $X$ among the integers is
$1-\frac{1}{\sqrt{3}}$.


Is this conjecture correct ?



A Conjecture on  the Density of a subset of integers
This is an update of my original response which was incorrect.


The conjecture is false. I will show that any $X$ in the conjecture has density at most


$$ 0.4226496 < 1-\frac{1}{\sqrt{3}}= 0.4226497...$$


Let $Y$ be the set of odd numbers whose prime exponents are odd and different from $9$. Clearly, any $X$ in the conjecture is a subset of $Y$. Yet, the density of $Y$ equals


$$ \frac{1}{2}\prod\_{p>2}\left(1-\frac{1}{p}\right)\left(1+\frac{1}{p-p^{-1}}-\frac{1}{p^9}\right) $$


which is less than


$$ 0.42264954363092750400907132916 $$


by the SAGE command



```
(1/2)*prod([RealField(100)(1-1/p)*(1+1/(p-1/p)-1/p^9) for p in prime_range(3,1000000)])

```

