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Homedetermining integer contraharmonic means

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For determining effectively values $c$ of integer contraharmonic means of two positive integers $u$ and $v$ ($1\,<\,u\,<\,v$), it’s convenient to start from the formula (7) in the parent entry:

$\displaystyle v\;=\;\frac{2u^{2}}{w}\!-\!u$ | (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

$\displaystyle c\;=\;\frac{2u^{2}}{w}\!-\!2u\!+\!w.$ | (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 formula for the integer harmonic mean, corresponding (2), is simply

$\displaystyle h\;=\;2u\!-\!w.$ | (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).

## Mathematics Subject Classification

11Z05*no label found*11A05

*no label found*11D09

*no label found*11D45

*no label found*

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