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 theparent 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 correspondingvalues 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.