I couldn't find any information on the subject in google, except for some archive reference desk, which offered some proof that I failed to understand. Basically what I am asking is, why aren't there other numbers n except from 18, for which sigma(n):n = 13:6, where sigma(n) is the sum of the divisors of n including n itself. — Preceding unsigned comment added by 130.204.34.208 (talk) 07:51, 13 July 2015 (UTC)[reply]

The proof here is based on decomposing n and looking at the size of ${\displaystyle \sigma (n)/n}$. If ${\displaystyle n=\prod _{i}p_{i}^{a_{i}}}$then ${\displaystyle \sigma (n)=\prod _{i}\sigma \left(p_{i}^{a_{i}}\right)=\prod _{i}{\frac {p_{i}^{a_{i}+1}-1}{p_{i}-1}}}$. Also if we let ${\displaystyle f(n)=\sigma (n)/n}$we have ${\displaystyle f(n)=\prod _{i}f\left(p_{i}^{a_{i}}\right)}$. All the factors are greater than 1. If you try ${\displaystyle n=2^{a}3^{b}m}$with ${\displaystyle a=2,\ b=1}$the product ${\displaystyle f(2^{a})f(3^{b})}$will already be greater than 13/6, and ${\displaystyle f(m)}$will just make it larger, so it can't be right. If you try ${\displaystyle a=1,\ b=2}$then the product will exceed 13/6 unless ${\displaystyle m=1}$and then you have 18. The rest of the proof follows, using also the facts that ${\displaystyle \sigma (6)=12}$, and that for a prime ${\displaystyle p>3}$we have ${\displaystyle \sigma (p^{a})}$is an integer and ${\displaystyle p^{a}}$is odd. -- Meni Rosenfeld (talk) 10:01, 13 July 2015 (UTC)[reply]

And, since I just can't get enough of Inside Out, I'll add that 18 is solitary because it lost the core memory that powers friendship island. -- Meni Rosenfeld (talk) 13:21, 14 July 2015 (UTC)[reply]

I suppose they are a step up from The Numskulls. That cartoon strip has survived for fifty years now. Dmcq (talk) 10:45, 17 July 2015 (UTC)[reply]