I just started Project Euler and I have run into an issue. There are numerous problems that ask about the proper divisors, sum of proper divisors, etc. of a number $N$. Now, before I proceed, I would like to make sure my vocabulary is correct. Is $N$ be considered a proper divisor of $N$? I have always said no, but since it does divide it, is the standard to include $N$ in a list of proper divisors?
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It is standard to exclude $N$ as a proper divisor. Of course, it is a divisor. The notion of proper divisor is little used in mathematics. For historical reasons, it crops up occasionally in problems connected with perfect numbers. Unless $N=1$, the other trivial divisor, namely $1$, is considered a proper divisor. – André Nicolas Feb 22 '14 at 07:12
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Thanks, that's what I thought, and yes, the Euler problem I'm working on is dealing with perfect numbers. – Milo Feb 22 '14 at 07:15
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It all comes from Classical and Hellenistic times, when an integer that properly divided $N$ was called something that translates to "part" (of $N$). The "perfection" of, for example $6$ is then the fact that $6$ is the sum of its parts. – André Nicolas Feb 22 '14 at 07:17
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No.
Indeed, the fact that $N$ is a divisor and not a proper divisor is the only difference between the words "divisor" and "proper divisor".
Eric Stucky
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