Call a fraction a "unit fraction" if it can be written as 1/n, where n is a positive integer.

How many more ways can the unit fraction 1/n be written as a sum of two (possibly equivalent) unit fractions than as a difference of two unit fractions?

The number of factors of n^2 is odd. the factors of n^2 run from 1 through n up to n^2 (where n is the middle factor). If a=x and b=y satisfy 1/n =1/a + 1/b , then so does a=y and b=x. To avoid repeated sums (where a and b values are reversed),

let b=n+f (where f represents a factor of n^2) and f runs from 1 to n.  Now # of sums = (F(n) + 1)/2.

I hope this clarifies how I got # of sums.

Posted by Dennis on 2006-10-17 13:20:23