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

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July 13

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Why is 18 a solitary number?

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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  . If   then  . Also if we let   we have  . All the factors are greater than 1. If you try   with   the product   will already be greater than 13/6, and   will just make it larger, so it can't be right. If you try   then the product will exceed 13/6 unless   and then you have 18. The rest of the proof follows, using also the facts that  , and that for a prime   we have   is an integer and   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]