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 σ1(n) of all their positive divisorsMathworldPlanetmathPlanetmath divides the sum σ2(n) of the squares of those divisors.  For example, 4 is such an integer:

1+2+4= 7 21= 12+22+42

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:// integer sequence

1, 4, 9, 16, 20, 25, 36, 49, 50, 64, 81, 100, 117, 121, 144, 169, 180,

Using the expressions of divisor functionDlmfDlmfMathworldPlanetmath ( σz(n), the condition for an integer n to be an antiharmonic number, is that the quotient


is an integer; here the pi’s are the distinct prime divisorsPlanetmathPlanetmath of n and mi’s their multiplicities.  The last form is simplified to

i=1kpimi+1+1pi+1. (1)

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


(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 (

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