## Monday, October 20, 2008

### An interesting Summation

(Guest post by Richard Matthew McCutchen.)

The formula below appears in The TeXbook as a typesetting exercise (I have slightly modified it): The expression has a clever mathematical meaning that is not discussed in the book. Try to find it and prove it!

Let p(n) be the limit as m &rarr &infin, m &isin Z, of

&sumk=0...&infin (1- cos{2m}(k!nπ/n))

(Comment from Bill: &pi is supposed to be pi, the ratio of circumference to diameter of a circle. html doesn't really do a good pi- if someone knows how to, in html, do a better one- let me know.)

1. Use LaTeX to make a GIF image. For example, from here:

http://thornahawk.unitedti.org/equationeditor/equationeditor.php

2. Why don't you simply type it in, like so: π?

3. It seems to me p(n) is the largest prime factor of n.

The sum is counting how many k's there are such that k!^n/n is not an integer, which happens iff some prime factor of n is larger than k.

4. kkmd: not quite. Suppose n is a large power of 2.

5. ignore the previous comment. I was forgetting the "^n"

Anyone know the value with the "^n" deleted?

6. For any fixed m, the infinite sum is actually finite since n divides k! for all k>=n. I have not looked at this carefully, but I think that p(n) is n minus the number of non-negative integers k less than n for which k!^n is a multiple of n.

7. By the same token, if you delet the ^n, you get the Kempner numbers :
http://www.research.att.com/~njas/sequences/A002034