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The antiharmonic, a.k.a. contraharmonic mean of some set of
positive numbers is defined as the sum of their squares
divided by their sum. There exist positive integers $n$
whose sum $\sigma_{1}(n)$ of all their positive divisors^{} divides
the sum $\sigma_{2}(n)$ of the squares of those divisors. For
example, 4 is such an integer:

$1+2+4\,=\,7\,\mid\,21\,=\,1^{2}+2^{2}+4^{2}$ |

Such integers are called antiharmonic numbers (or contraharmonic numbers), since the contraharmonic mean of their positive divisors is an integer.

The antiharmonic numbers form the OEIS integer sequence A020487:

$1,\,4,\,9,\,16,\,20,\,25,\,36,\,49,\,50,\,64,\,81,\,100,\,117,\,121,\,144,\,16% 9,\,180,\,\ldots$ |

Using the expressions of divisor function $\sigma_{z}(n)$, the condition for an integer $n$ to be an antiharmonic number, is that the quotient

$\sigma_{2}(n):\sigma_{1}(n)\;=\;\sum_{{0<d\mid n}}\!d^{2}:\!\sum_{{0<d\mid n}}% \!d\;=\;\prod_{{i=1}}^{k}\frac{p_{i}^{{2(m_{i}+1)}}-1}{p_{i}^{2}-1}:\prod_{{i=% 1}}^{k}\frac{p_{i}^{{m_{i}+1}}-1}{p_{i}-1}$ |

is an integer; here the $p_{i}$’s are the distinct prime divisors^{}
of $n$ and $m_{i}$’s their multiplicities. The last form is
simplified to

$\displaystyle\prod_{{i=1}}^{k}\frac{p_{i}^{{m_{i}+1}}+1}{p_{i}+1}.$ | (1) |

The OEIS sequence A020487 contains all nonzero perfect squares, since in the case of such numbers the antiharmonic mean (1) of the divisors has the form

$\prod_{{i=1}}^{k}\frac{p_{i}^{{2m_{i}+1}}+1}{p_{i}+1}\;=\;\prod_{{i=1}}^{k}% \left(p_{i}^{{2m_{i}}}-p_{i}^{{2m_{i}-1}}-\!+\ldots-p_{i}+1\right)$ |

Note. It would in a manner be legitimated to define a positive integer to be an antiharmonic number (or an antiharmonic integer) if it is the antiharmonic mean of two distinct positive integers; see integer contraharmonic mean and contraharmonic Diophantine equation.

## Mathematics Subject Classification

11A05*no label found*11A25

*no label found*

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## Comments

## a new article invisible

Hi admins! My new article ”antiharmonic number” turned invisible. When I deleted it, my score increased(!) 100 points. Then I wrote the article anew, the score increased again 100 points and the article is again invisible. There must be some system bug. Jussi

## make test entry and delete

You earn 200 points!

## gratis PM author points

Hi admins, making a new entry gives 100 points, deleting it also gives 100 points!! I have lately deleted one of my entries (getting 100 p) and made two test entries and deleted them (thus getting 400 p).

Please correct this system bug and subtract 500 points from me!

Jussi

## re: gratis PM author points - oops

I’ll look into that shortly, thanks for the alert!

Joe

## re: gratis PM author points - fixed

Jussi: it was the old classic, ”missing a minus sign”. Fixed now!

Thanks kindly!

Joe

## re: re: gratis PM author points -- fixed

Thanks Joe, it’s very good! I also have tested it. But please subtract the 500 gratis points of me!

Best wishes, Jussi

## Re: a new article invisible

I don’t know about the deletion, but I can say why it is invisible. At the end of the second paragraph, the equation starts with a double dollar sign but ends with a single dollar sign. This is the TeX bug which is causing it not to render. Unfortuinately, as it stands now, when it runs across such a bug and cannot render the article, LaTeXML simply produces no output whatsoever rather than producing an error message which could help the user fix the problem or at least be made aware of what happened.

## Thank you very much, rspuzio!

Thank you very much, rspuzio! You were right; I did’nt see the lacking dollar sign since I have only a 14”-monitor =o(

Regards, Jussi