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?


$12285$ 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.

  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.