determining integer contraharmonic means

For determining effectively values $c$ of integer contraharmonic means of two positive integers $u$ and $v$ (), it’s convenient to start from the (7) in the parent entry (http://planetmath.org/IntegerContraharmonicMeans):

$v={\displaystyle \frac{2u^2}{w}}-u$ (1)

where $w$ is any positive factor of $2u^2$ less than $u$.  Substituting the above expression of $v$ to the defining expression

$$c=\frac{u^2+v^2}{u+v}$$

of $c$, this gets the form

$c={\displaystyle \frac{2u^2}{w}}-2u+w.$ (2)

Hence one can use the formulae (1) and (2), giving in them for each desired $u$ the values $w$ of the positive factors of $2u^2$, beginning from  $w:=1$  and stopping before  $w=u$.

The for the integer harmonic mean, corresponding (2), is simply

$h=\mathrm{\hspace{0.33em}2}u-w.$ (3)

Example.  In the following table one sees for  $u=36$  all possible values of the parametre $w$ and the corresponding values of $c$ and $h$; the pertinent values of $v$ are given, too.

$w$	$1$	$2$	$3$	$4$	$6$	$8$	$9$	$12$	$16$	$18$	$24$	$27$	$32$
$v$	$2556$	$1260$	$828$	$612$	$396$	$288$	$252$	$180$	$126$	$108$	$72$	$60$	$45$
$c$	$2521$	$1226$	$795$	$580$	$366$	$260$	$225$	$156$	$106$	$90$	$60$	$51$	$41$
$h$	$71$	$70$	$69$	$68$	$66$	$64$	$63$	$60$	$56$	$54$	$48$	$45$	$40$

As one sees, the contraharmonic and the harmonic mean may differ considerably, but also the difference 1 is possible.

Title	determining integer contraharmonic means
Canonical name	DeterminingIntegerContraharmonicMeans
Date of creation	2013-11-19 18:13:25
Last modified on	2013-11-19 18:13:25
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Algorithm
Classification	msc 11Z05
Classification	msc 11A05
Classification	msc 11D09
Classification	msc 11D45
Related topic	LinearFormulasForPythagoreanTriples