antiharmonic number

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

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

Such integers are calledantiharmonic numbers (or contraharmonic numbers),since the contraharmonic mean of their positive divisors is aninteger.

The antiharmonic numbers form theHTTP://oeis.org/OEIS integer sequencehttp://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 aninteger $n$ 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 divisorsof $n$ and $m_{i}$ ’s their multiplicities. The last form issimplified 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) ofthe 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 definea positive integer to be an antiharmonic number (or anantiharmonic integer) if it is the antiharmonic mean of twodistinct positive integers; see integer contraharmonic meanand contraharmonic Diophantine equation (http://planetmath.org/ContraharmonicDiophantineEquation).