Wikipedia:Reference desk/Archives/Mathematics/2015 July 13

= July 13 =

Why is 18 a solitary number?
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)
 * The proof here is based on decomposing n and looking at the size of $$\sigma(n)/n$$. If $$n=\prod_ip_i^{a_i}$$ then $$\sigma(n)=\prod_i\sigma\left(p_i^{a_i}\right)=\prod_i\frac{p_i^{a_i+1}-1}{p_i-1}$$. Also if we let $$f(n)=\sigma(n)/n$$ we have $$f(n)=\prod_if\left(p_i^{a_i}\right)$$. All the factors are greater than 1. If you try $$n=2^a3^bm$$ with $$a=2,\ b=1$$ the product $$f(2^a)f(3^b)$$ will already be greater than 13/6, and $$f(m)$$ will just make it larger, so it can't be right. If you try $$a=1,\ b=2$$ then the product will exceed 13/6 unless $$m=1$$ and then you have 18. The rest of the proof follows, using also the facts that $$\sigma(6)=12$$, and that for a prime $$p>3$$ we have $$\sigma(p^a)$$ is an integer and $$p^a$$ is odd. -- Meni Rosenfeld (talk) 10:01, 13 July 2015 (UTC)
 * 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)
 * 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)