Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Question

Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$.

$\{y, z\}$ is said to be a friendly pair if

$$I(y) = I(z),$$

where $I(x) = \sigma(x)/x$ is the abundancy index of $x$.

As an example, note that $\{30, 140\}$ is a friendly pair, because $30$ and $140$ have the same abundancy:

$$I(30) = \frac{\sigma(30)}{30} = \frac{1 + 2 + 3 + 5 + 6 + 10 + 15 + 30}{30} = \frac{72}{30} = \frac{12}{5}$$

$$I(140) = \frac{\sigma(140)}{140} = \frac{1 + 2 + 4 + 5 + 7 + 10 + 14 + 20 + 28 + 35 + 70 + 140}{140} = \frac{336}{140} = \frac{12}{5}$$

Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Answer 1

$12285$ and $14595$ are an odd couple; there are others known.

Remark: I believe it is not known whether there is a "friendly pair" where one is odd and the other is even. Since the Pythagoreans considered even numbers to be female, and odd numbers male, this may be significant.

Added: For the sake of the joke, I am leaving the remark. However, please see the comment by Jose Arnaldo Dris below.