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I came across the definition of a highly composite number yesterday as a positive integer that has more divisors than any positive integer smaller than it. And, then I realised it would give 2 a very awkward distinction of being the only number that's both prime and highly composite. Is this correct ?

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    $\begingroup$ Sometimes life is stranger than fiction :) $\endgroup$ Jul 8, 2016 at 5:56
  • $\begingroup$ Of course $2$ is highly composite... It has more divisors than $1$ does :-) $\endgroup$
    – Frenzy Li
    Jul 8, 2016 at 5:56
  • $\begingroup$ @StevenGregory What is the relation between division by zero being undefined and my question ? $\endgroup$
    – Saikat
    Jul 8, 2016 at 6:00
  • $\begingroup$ No-ones dividing by 0. ???? Yes, it does indeed seem to be that 2 is indeed unique in being prove prime and highly composite. I think that's kind of neat. $\endgroup$
    – fleablood
    Jul 8, 2016 at 6:03
  • $\begingroup$ For the definition to be meaningful you have to restrict the set of numbers to strictly positive integers. Otherwise no positive integer will be highly composite. $\endgroup$
    – skyking
    Jul 8, 2016 at 6:19

2 Answers 2

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If you think of a highly composite number as any number that is "more composite" than any smaller number, it would make sense that the very first highly composite number wouldn't be composite at all.

Actually according to the wikipedia page 1 is a highly composite number! With 0 prime factors.

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  • $\begingroup$ "Most composite thus far number" is too long... :) $\endgroup$
    – Frenzy Li
    Jul 8, 2016 at 6:23
  • $\begingroup$ I appreciate your posts on my last few threads, Fleablood. By the way, why is that your username ? Also are you also an undergraduate student ? $\endgroup$
    – Saikat
    Jul 8, 2016 at 7:05
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I suppose you could require that a highly composite number actually be composite, but it's never going to happen for any other prime, as they don't have very many divisors at all.....

But yes, this is true strictly from the definition you gave.

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