Date: Nov 23, 2012 7:16 AM
Author: Bill Taylor
Subject: Re: Reciprocals of integers summing to 1

On Nov 23, 3:08 pm, david petry <david_lawrence_pe...@yahoo.com>
wrote:
> On Thursday, November 22, 2012 4:07:09 AM UTC-8, 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
>
> > It seems that  #(p) = 2  for prime p, and in general

In fact this is simply routinely proved, unsurprisingly.

> > 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.

Of course!  Neat triviality.     I wonder if the surplus ways
are also simply characterizable.   Must investigate.

-- Bumbling Bill

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