
Re: Reciprocals of integers summing to 1
Posted:
Nov 22, 2012 9:08 PM


On Thursday, November 22, 2012 4:07:09 AM UTC8, Bill Taylor wrote: > > If anyone wants to check my hand work, the number > > of ways of splitting 1/m is (I suggest) > > > > m: 1 2 3 4 5 6 7 8 9 10 12 15 18 20 24 42 > > #(m) 1 2 2 3 2 5 2 4 3 5 4 5 8 8 11 14 > > > > (the gaps are where I didn't need them for the above work). > > > > This table at least is easy to do by computer. > > It seems that #(p) = 2 for prime p, and in general > > the # function bears a close resemblance to d, the number > > of factors of m; though it tends to be a bit bigger, > > and is not multiplicative like d.
The formula 1/(a*b) = 1/(a*(a+b)) + 1/(b*(a+b)) shows why there is a close relation between your function and the number of ways the number can be factored into two factors.

