Density of numbers which are the product of distinct primes raised to prime powers Consider the sequence of natural numbers which are the product of distinct primes raised to prime powers (http://oeis.org/A056166) The first few numbers in this sequence are
$$
4, 8, 9, 25, 27, 32, 36, 49, 72, 100, 108, 121, 125, 128, 169, 196, 200, 216, 225, 243, 288, \ldots
$$
Question: Let $f(x)$ be the number of such numbers $\le x$. Experimental data for $x \le 4 \times 10^9$ shows that $f(x) \sim a \sqrt x$ some constant $a$. Can this is proved or disproved?
Also the for this range of data the computed value of the parameter $a$ is approximately $1.416$ which is pretty close to $\sqrt 2$.
Plot of $f(x)$


If you let $a_n = 1$ if $n$ has this property and $a_n = 0$ otherwise, then
$$F(s) = \sum \frac{a_n}{n^s} = \prod_{p} \left(1 + \frac{1}{p^{2s}} + \frac{1}{p^{3s}} + \frac{1}{p^{5s}} + \ldots \right),$$
On the other hand,
$$\zeta(2s) = \prod_{p} \left(1 - \frac{1}{p^{2s}}\right)^{-1},$$
and thus
$$\frac{F(s)}{\zeta(2s)} = \prod_{p} \left(1 + \frac{1}{p^{3s}} - \frac{1}{p^{4s}} - \frac{1}{p^{9s}} + \ldots \right),$$
where the exponents are those occuring in $(1-x^2) \sum x^p = 1 + x^3 - x^4 - x^9 + \ldots$.
From this, you see that $F(s)$ is holomorphic up to $s = 1/2$ where there is a simple pole with residue $C/2$, where
$$C =  \prod_{p} \left(1 + \frac{1}{p^{3/2}} - \frac{1}{p^{2}} - \frac{1}{p^{9/2}} + \ldots \right),$$
$$ = 1.4310606003 \ldots $$
But then, the Wiener–Ikehara theorem (and its natural variants)
$$\sum_{n<X} a_n \sim C x^{1/2} + O(x^{1/3}).$$
