determining integer contraharmonic means
For determining effectively values $c$ of integer contraharmonic means of two positive integers $u$ and $v$ ($$), it’s convenient to start from the (7) in the parent entry (http://planetmath.org/IntegerContraharmonicMeans):
$v={\displaystyle \frac{2{u}^{2}}{w}}-u$ | (1) |
where $w$ is any positive factor of $2{u}^{2}$ less than $u$. Substituting the above expression of $v$ to the defining expression
$$c=\frac{{u}^{2}+{v}^{2}}{u+v}$$ |
of $c$, this gets the form
$c={\displaystyle \frac{2{u}^{2}}{w}}-2u+w.$ | (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 $2{u}^{2}$, 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.
References
- 1 J. Pahikkala: “On contraharmonic mean and Pythagorean triples^{}”. – Elemente der Mathematik 65:2 (2010).
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 |