For , define oto be the highest exponent to which a prime divides it. For instance, . Prove that exists.

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- June 12th 2009, 06:42 PMChandru1exponents
For , define oto be the highest exponent to which a prime divides it. For instance, . Prove that exists.

- June 13th 2009, 06:05 AMPaulRS
**Previous Result:**The cardinal of: is where is the floor function and is the Möbius function.

*Proof*

Consider: , the cardinal of this set is, by inclusion-exclusion:

But: thus note that we can write: but we want and the rest follows.

At this point, note that every number that is free of squares, is free of cubes ... and so on. Further, if a number is free of cubes, but not of squares, then

That is:

Thus we get: (This is already very suggestive)

Mmmm, here I've not been able to finish it formally -it gets quite nasty-, but I do think that: -remember and -