(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.)
Use LaTeX to make a GIF image. For example, from here:
ReplyDeletehttp://thornahawk.unitedti.org/equationeditor/equationeditor.php
Why don't you simply type it in, like so: π?
ReplyDeleteIt seems to me p(n) is the largest prime factor of n.
ReplyDeleteThe 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.
kkmd: not quite. Suppose n is a large power of 2.
ReplyDeleteignore the previous comment. I was forgetting the "^n"
ReplyDeleteAnyone know the value with the "^n" deleted?
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.
ReplyDeleteBy the same token, if you delet the ^n, you get the Kempner numbers :
ReplyDeletehttp://www.research.att.com/~njas/sequences/A002034