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A perfect number is an integer $n$ greater than $1$ that equals the sum of its factors, excluding $n$ itself. For example, $6 = 1 + 2 +3 $ so $6$ is perfect. It is unknown whether there are any odd perfect numbers. My question is, are there any odd integers $n$ greater than $1$ such that the sum of all of $n$'s factors, excluding $n$, is greater than $n$? In number-theoretic language, does there exist odd $n$ with $\sigma(n) > 2n$?

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Shouldn't your title say "less than or equal to"? –  Grumpy Parsnip May 9 '12 at 22:54
    
No. As I said, it is well-known that it is unknown whether there are any odd perfect numbers. –  Stefan Smith May 9 '12 at 23:00
    
@Jim: I think he means proper divisors –  The Chaz 2.0 May 9 '12 at 23:01
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@user20520: my point is that if the answer to your question as stated in the title was "yes", then you would have solved a famous unsolved problem, so you probably want to include the possibility of equality. –  Grumpy Parsnip May 9 '12 at 23:02
    
No, I have not solved that famous problem! It just seemed unlikely that any odd number could be less than the sum of its proper divisors, if you think about small numbers like $15$, $35$, etc. See the answer below - the smallest odd number with the desired property is $945$ –  Stefan Smith May 9 '12 at 23:05

2 Answers 2

up vote 4 down vote accepted

Look up "abundant numbers" in Wikipedia and at https://oeis.org/A005101

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The odd abundant numbers have an entry of their own. –  hardmath May 9 '12 at 23:06
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Actually A006038 are the odd primitive abundant numbers. The odd abundant numbers are oeis.org/A005231 –  Robert Israel May 9 '12 at 23:18

Indeed. The smallest one is $945$.

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Did you know that offhand? –  Alex Becker May 9 '12 at 22:58
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@AlexBecker: Yes. Unfortunately, given boundedness of my memory, it just means that there are a lot more important things I do not have space for. –  André Nicolas May 9 '12 at 22:59
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@André Nicolas: Just because your memory is bounded does not mean you have reached its limit. I'll wager that your memory is at most 1-1/e full. –  marty cohen May 10 '12 at 2:11
    
Can you come up with solution by using number theory ? –  Frank Jan 26 '13 at 19:37
    
Yes, and getting that $945$ is the first is easy, but identifying say the $10$-th involves some fiddling. Start from the standard expression for $\frac{\sigma(n)}{n}$ in terms of the prime factorization. We want to get past $2$. Of course we will use the first few odd primes, and the exponents must be non-increasing. We have to weigh increasing the exponent of a small prime versus adding a new prime. Sometimes it can be close. –  André Nicolas Jan 26 '13 at 19:44

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