An Identity Concerning Averages
of Divisors of a Given Integer

Let me start with two definitions. Given a sequence of N > 0 numbers a₁, a₂, ..., a_N, we define an arithmetic mean or average is defined

A(a1, a2, ..., aN) = (a1 + a2 + ... + aN)/N

If none of the numbers is 0, we also define their harmonic mean by

H(a1,a2,...,aN) = N/(1/a1 + 1/a2 + ... + 1/aN)

Let there be an integer M. We may consider all possible divisors of M, including 1 and M itself. We do not know how many there are. Nonetheless, for a given M, we may consider the arithmetic and harmonic means of the set of divisors. I denote them A_M and H_M, respectively. The task of computing either, especially the harmonic mean, may seem daunting. Thus it's all the more amusing to have the following

Theorem

M = A_M · H_M.

Examples

1. M = 6 has four divisors: 1, 2, 3,and 6. A₆ = (1 + 2 + 3 + 6)/4 = 3. On the other hand,

   H6 = 4/(1/1 + 1/2 + 1/3 + 1/6) = 4/(6/6 + 3/6 + 2/6 + 1/6) = 24/12 = 2.

   Therefore we indeed have 6 = 3·2.

2. M = 8 has also four divisors: 1,2,4, and 8.

   A8 = (1 + 2 + 4 + 8)/4 = 15/4. H8 = 4/(1/1 + 1/2 + 1/4 + 1/8) = 4/(8/8 + 4/8 + 2/8 + 1/8) = 32/15.

   Finally, 15/4·32/15 = 32/4 = 8.

Proof

The main point here is to realize that the divisors of a number come in pairs: d divides M iff there exists an integer c such that M/d = c. This also can be written as

(1)

1/d = c/M

Now, let's enumerate all the divisors of M as d₁, d₂,... We know that for every d_i there exists a c_i such that d_i·c_i = M for all i. Please note that c_i is, by definition, also a divisor of M. Furthermore, as d_i ranges over the set of all divisors of M, so does c_i. Let D_M denote the number of divisors of M. From (1) we get

HM = DM/(1/d1 + 1/d2 + ...) = DM/(c1/M + c2/M + ...) = M/AM.

Which exactly means that M = A_M · H_M.

References

1. O.Ore Number Theory and Its History, Dover, 1988

Related material Read more...
	The Means
	Averages, Arithmetic and Harmonic Means
	Expectation
	The Size of a Class: Two Viewpoints
	Family Statistics: an Interactive Gadget
	Averages in a sequence
	Arithmetic and Geometric Means
	Geometric Meaning of the Geometric Mean
	A Mathematical Rabbit out of an Algebraic Hat
	AM-GM Inequality
	The Mean Property of the Mean
	Harmonic Mean in Geometry

|Contact| |Front page| |Contents| |Up|

Copyright © 1996-2018 Alexander Bogomolny