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I know very well what is a superior highly composite number, but I would like to see how could we (roughly) define what is a superior highly composite number using only words (using no equations and no inequalities).

The only such definition I've seen is this one found on Wikipedia:

A superior highly composite number is a natural number which has more divisors than any other number scaled relative to the number itself.

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    $\begingroup$ If it wasn't for the exponent $\epsilon$ you could say "A superior highly composite number is a number which is divided by a greater fraction of the numbers less than or equal to it than any smaller number." But the exponent seems to defeat any attempts at clear phrasing. $\endgroup$ – hobbs Sep 6 '14 at 23:32
  • $\begingroup$ Their definition isn't a very good solution of the exponent problem. I think the problem with using no symbols at all is that it's nearly impossible to convey that we're speaking about a single constant $\epsilon$ for each $n$ and all $k<n$, rather than a different constant for each $k$. $\endgroup$ – Jaycob Coleman Feb 15 '15 at 22:43
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Natural numbers were involved in countable number of competitions.

The score in an individual competition was calculated as the ratio of the number of divisors of the competing number to the ...

  • in the $1^{st}$  competition:   ... to the competing number itself
  • in the $2^{nd}$ competition:   ... to the square of the competing number
  • in the $3^{rd}$  competition:  ... to the cube of the competing number
  •     ... and so on ...

The absolute winners of the individual competitions obtained the title Superior highly composite number.

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  • $\begingroup$ wait what about the competitions where the competing numbers are taken to the exponent of a non-whole number for example $n^{1.5}$ or $n^{3.27913452218237...}$ etc. $\endgroup$ – ray lin Jun 16 '18 at 20:32

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