antiharmonic number

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 HTTP://oeis.org/OEIS integer sequence http://oeis.org/search?q=A020487&language=english&go=SearchA020487:

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 (http://planetmath.org/DivisorFunction) $\sigma_{z}(n)$ , 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)

(cf. irreducibility of binomials with unity coefficients).

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 (http://planetmath.org/ContraharmonicDiophantineEquation).

Title	antiharmonic number
Canonical name	AntiharmonicNumber
Date of creation	2013-11-28 10:15:29
Last modified on	2013-11-28 10:15:29
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Definition
Classification	msc 11A05
Classification	msc 11A25