3. Among the natural numbers from 1 to 144, the number of ways to pick three numbers that form an increasing geometric progression with an integer common ratio is $\qquad$ .
3. 78.

Let the three numbers $a$, $a q$, and $a q^{2}$ form an increasing geometric sequence. Then
$$
1 \leqslant a < a q < a q^{2} \leqslant 144 \text{.}
$$

From this, we have $2 \leqslant q \leqslant 12$.
When $q$ is fixed, the number of integers $a$ such that the three numbers $a$, $a q$, and $a q^{2}$ are integers is denoted as $N(q)$.
$$
\begin{array}{l}
\text{By } a q^{2} \leqslant 144 \text{, we have } N(q) = \left[\frac{144}{q^{2}}\right] \text{.} \\
\text{Also, } N(2) = 36, N(3) = 16, N(4) = 9, \\
N(5) = 5, N(6) = 4, N(7) = N(8) = 2, \\
N(9) = N(10) = N(11) = N(12) = 1 \text{.}
\end{array}
$$

Therefore, the total number of ways is
$$
N(2) + N(3) + \cdots + N(12) = 78 \text{ (ways).}
$$