TITLE: Is it possible to cover $\{1,2,...,100\}$ with $20$ geometric progressions?
QUESTION [23 upvotes]: Recall that a sequence $A=(a_n)_{n\ge 1}$ of real numbers is said to be a geometric progression whenever $\dfrac{a_{n+1}}{a_n}$ is constant for each $n\ge 1$. Then, replacing $20$ with $12$, the following question comes from an old Russian competition:

Problem: Can you find $20$ geometric progressions $A_1,A_2,\ldots,A_{20}$ of real numbers such that
$$
\{1,2,\ldots,100\} \subseteq A_1 \cup A_2\cup \ldots \cup A_{20}?
$$

From this question, it is known that we can cover $\{1,2,\ldots,100\}$ with $36$ geometric progressions.  In fact, $36$ is the least possible number.

REPLY [14 votes]: Any geometric progression of reals contains at most two squarefree integers.
Since there are $61$ squarefree integers in $\{1, 2, 3, \ldots, 100\}$ (see OEIS A005117 and A013928) (note $1$ is considered squarefree), at least $\left\lceil{\frac{61}{2}}\right\rceil = 31$ geometric progressions are required. In particular you cannot do it with only $20$.

Slightly longer answer:
If $\mathcal{G}$ is a geometric progression of positive real numbers, then $\mathcal{G} \cap \mathbb{Q}$ is a geometric progression of positive rational numbers, and $\mathcal{G} \cap \mathbb{Z}$ is a geometric progression of positive integers (though the ratio may be a non-integer rational). So we only need to consider positive integer geometric progressions. Suppose the common ratio is $\frac{r}{s}$, where $r,s$ are relatively prime integers and $r > s \ge 1$.
The geometric progression may be finite or infinite, but it has a minimum element. If $a$ is that minimum element, then the terms can be listed out:
$$
a, \frac{ar}{s}, \frac{ar^2}{s^2}, \frac{ar^3}{s^3}, \cdots.
$$
We see that of the terms $\frac{ar^k}{s^k}$ for $k \ge 2$ are divisible by $r^2$ (since $r$ and $s$ are relatively prime), and since $r \ge 2$, they are not squarefree. So only $a$ and $\frac{ar}{s}$, the first two terms, can potentially be squarefree.
Therefore, there are at most $2$ squarefree integers in the geometric progression.