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Topic: The Largest Versatile Number
Replies: 13   Last Post: Jul 9, 1996 12:31 PM

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 Lance Levine Posts: 6 Registered: 12/12/04
Re: The Largest Versatile Number
Posted: Jun 29, 1996 2:09 PM

>Versatile Numbers are numbers that have more factors than any smaller
>number.
>Twelve is a versatile number because 12 has 6 factors and no number less
>than
>12 has that many factors. These are the first few versatile numbers: 2,
>4,
>6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 840, 1260, 1680, 2520, 5040,
>...

I have also been fooling around with this set of numbers, and, strangly enough,
I also labeled them "versatile numbers" :) But my definition of a versatile
number is different from yours:

Let f(n)=the sum of the factors of n
V(n) --the versatility of n-- = f(n)/n.

I believe both definitions lead to the same set of numbers, because your list
is equivalent to mine, at least for the first 40 terms, which is all I have
calculated... Prime numbers have V(n)=(n+1)/n, and the 43rd versatile number
(the largest one I've found--122522400), has V(n) slightly above 5.

It's quite maddenning to try and search for patterns in the
versatile numbers; every time you think you've spotted one, it turns out not
to work. One interesting pattern, though, is that the first two versatile
numbers are powers of 2, then a factor of 3 is added, then later a factor of
5, 7, 11, etc.