A Conjecture on  the Density of a subset of integers 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: Let $A$ be any set of positive integers such that $1\in A$, and let $B$ be the set of integers such that every exponent in the prime factorization of any $x\in B$ belongs to $A$. Then the density of $B$ exists and equals
$$
\prod_{\text{primes }p} \bigg( 1-\frac1p \bigg) \bigg( 1 + \sum_{a\in A} \frac1{p^a} \bigg).
$$
In this case, $A=B=X$ (which is kind of cool).
A: 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)])

