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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?

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$12285$ 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.

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    $\begingroup$ In the following document titled "Western Number Theory Problems, 12, 16 & 18 Dec 2008" and edited by Gerry Myerson, two examples of such "friendly pairs" are given where one is odd and the other is even: $$n = 42, m = 544635 = {3^2}\cdot{5}\cdot{7^2}\cdot{13}\cdot{19},$$ and $$n = {2}\cdot{3^6}\cdot{23}\cdot{137}\cdot{547}\cdot{1093}, m = {3^4}\cdot{5}\cdot{7}\cdot{11^2}\cdot{19}.$$ $\endgroup$ Oct 28 '13 at 7:06
  • $\begingroup$ Per the same document: Many more examples can be extracted from upforthecount.com/math/ffp8.html $\endgroup$ Oct 29 '13 at 9:26
  • $\begingroup$ To save the joke, it is unknown if "female" and "male" can be amicable (which is something else than friendly). $\endgroup$ Dec 26 '13 at 23:31
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    $\begingroup$ Extending my older comment. The numbers you mention, $12285$ and $14595$ are not (mutually) friendly. For $I(12285)=\frac{256}{117}\ne\frac{256}{139}=I(14595)$ where $I$ is as in the question. Instead, this is an amicable pair, since $\sigma(12285)-12285=14595$ and $\sigma(14595)-14595=12285$. To find a pair of friendly, odd numbers is not hard. For example $135$ and $819$ will do since $I(135)=\frac{16}{9}=I(819)$. $\endgroup$ Sep 11 '16 at 22:23
  • $\begingroup$ @JeppeStigNielsen: Sorry, still recovering, will not be able to respond for a few days. $\endgroup$ Sep 23 '16 at 23:36
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Since the old answer is confusing friendly numbers and amicable numbers, and since A. Nicolas may not edit it, I might as well supply a new answer.

$135$ and $819$ are a friendly couple of odd numbers. It is not hard to find more cases by brute force.

For a case of a "heterophile" friendship (by which I mean a friendly pair consisting of one even and one odd number), consider e.g. $42$ and $544635$. This example was already mentioned by the asker in a comment to the old answer.

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