# 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:

$$1+2+4=\mathrm{\hspace{0.17em}7}\mid \mathrm{\hspace{0.17em}21}={\mathrm{\hspace{0.17em}1}}^{2}+{2}^{2}+{4}^{2}$$ |

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:

$$1,\mathrm{\hspace{0.17em}4},\mathrm{\hspace{0.17em}9},\mathrm{\hspace{0.17em}16},\mathrm{\hspace{0.17em}20},\mathrm{\hspace{0.17em}25},\mathrm{\hspace{0.17em}36},\mathrm{\hspace{0.17em}49},\mathrm{\hspace{0.17em}50},\mathrm{\hspace{0.17em}64},\mathrm{\hspace{0.17em}81},\mathrm{\hspace{0.17em}100},\mathrm{\hspace{0.17em}117},\mathrm{\hspace{0.17em}121},\mathrm{\hspace{0.17em}144},\mathrm{\hspace{0.17em}169},\mathrm{\hspace{0.17em}180},\mathrm{\dots}$$ |

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

$$ |

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

$\prod _{i=1}^{k}}{\displaystyle \frac{{p}_{i}^{{m}_{i}+1}+1}{{p}_{i}+1}}.$ | (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

$$\prod _{i=1}^{k}\frac{{p}_{i}^{2{m}_{i}+1}+1}{{p}_{i}+1}=\prod _{i=1}^{k}({p}_{i}^{2{m}_{i}}-{p}_{i}^{2{m}_{i}-1}-+\mathrm{\dots}-{p}_{i}+1)$$ |

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 |