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 $\sigma(n) > 2n$?
Is any odd natural number less than the sum of its factors?
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elementary-number-theory
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0No, 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 $945$ – 2012-05-09
2 Answers
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Look up "abundant numbers" in Wikipedia and at https://oeis.org/A005101
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1Actually A006038 are the odd *primitive* abundant numbers. The odd abundant numbers are https://oeis.org/A005231 – 2012-05-09
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Indeed. The smallest one is $945$.
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0Oh I couldn't follow that.Would you please clarify more and add it to the answer ? – 2013-01-28