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)
$$\sum_{n<X} a_n \sim C x^{1/2} + O(x^{1/3}).$$