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So, I wondered if the property described in the title (namely, the property that the sum of the divisors of $n$ equals the sum of the divisors of $n+1$) ever occurred, and went to compute it. Here are the numbers with this property up to 20.000 (including):

14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, 19358, ...

Can anyone explain this growth? Are there infinitely many of them? (sure looks like so). Is there a formula for the nth term of this sequence, or something?

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I've edited to incorporate the title into the body, to make the question self-contained. –  Gerry Myerson Aug 20 '11 at 23:28
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up vote 5 down vote accepted

Whether or not this sequence goes on forever is an unsolved problem. You can find its terms and more information at the OEIS. It is A002961.

Edit: these are the numbers for which $n$ and $n+1$ have the same sum of divisors.

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It seems that the numbers you listed are squarefree numbers or numbers of the form $p^{k}q$, where $p$ is the smallest prime factor of such a number and $q$ a squarefree number.

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The OEIS page has a link to the first 4800 numbers with $\sigma(n)=\sigma(n+1)$. I'd be more impressed if you checked more than just the 12 numbers OP listed. –  Gerry Myerson Aug 21 '11 at 12:57
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My previous observation is false: 193893=3.7².1319. –  Sylvain Julien Aug 21 '11 at 14:51
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