Divisor Function - Properties

Properties

For a non-square integer every divisor d of n is paired with divisor n/d of n and is then even; for a square integer one divisor (namely ) is not paired with a distinct divisor and is then odd.

For a prime number p,

because by definition, the factors of a prime number are 1 and itself. Also,where p_n# denotes the primorial,

since n prime factors allow a sequence of binary selection ( or 1) from n terms for each proper divisor formed.

Clearly, 1 < d(n) < n and σ(n) > n for all n > 2.

The divisor function is multiplicative, but not completely multiplicative. The consequence of this is that, if we write

where r = ω(n) is the number of distinct prime factors of n, p_i is the ith prime factor, and a_i is the maximum power of p_i by which n is divisible, then we have

which is equivalent to the useful formula:

It follows (by setting x = 0) that d(n) is:

For example, if n is 24, there are two prime factors (p₁ is 2; p₂ is 3); noting that 24 is the product of 23×31, a₁ is 3 and a₂ is 1. Thus we can calculate d(24) as so:

The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.

We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, i.e. the divisors of n excluding n itself. This function is the one used to recognize perfect numbers which are the n for which s(n) = n. If s(n) > n then n is an abundant number and if s(n) < n then n is a deficient number.

If n is a power of 2, e.g., then and s(n) = n - 1, which makes n almost-perfect.

As an example, for two distinct primes p and q with p < q, let

Then

and

where φ(n) is Euler's totient function.

Then, the roots of:

allows us to express p and q in terms of σ(n) and φ(n) only, without even knowing n or p+q, as:

Also, knowing n and either σ(n) or φ(n) (or knowing p+q and either σ(n) or φ(n)) allows us to easily find p and q.

In 1984, Roger Heath-Brown proved that

d(n) = d(n + 1)

will occur infinitely often.