The definition and key theorems involving Euler's Totient function.  For more info, see the "related pages" listed below.

Definition

τ(n) is the number of positive divisors of n

Calculating τ(n)

Let x = p₁^e1 p₂^e2 ... p_n^en 

The factors of x are all the numbers of the form

y = p₁^d1 p₂^d2 ... p_n^dn,

where the d_i each independently range from 0 to e_i.  There are (1+e_i) ways to choose the exponent of the i'th prime factor, so

τ(x) = (1+e₁) (1+e₂)...(1+e_n)

A surprising identity: sum(k=1 to n) tau(k) = sum(h=1 to n) [n/h]

n Σ k=1

τ(k)  =

n Σ h=1

[n/h]

This is most easily proved by induction on n.  It's true when n=1, because τ(1) = 1, and [1/1] = 1.

[(n+1)/h] - [n/h] = 1 or 0, depending on whether h is a divisor of n+1 or not.  Therefore,

n+1 Σ h=1

[(n+1)/h]  −

n+1 Σ h=1

[n/h] = τ(n+1)

And since [n/(n+1)]=0,

n+1 Σ h=1

[(n+1)/h]  −

n Σ h=1

[n/h] = τ(n+1),

which establishes the truth of the identity for n+1.

Related pages in this website

φ(n), the number of coprimes to n: Totient function 

μ(n), the M�bius function 

The webmaster and author of this Math Help site is Graeme McRae.