## 5.11.10

### A Function-Theoretic Analysis of the Abundancy Index

A.   Analysis of the Abundancy Index

Let f(p, k) = I(p^k) = Sigma[p^k]/(p^k).

This is a rational function of two parameters: a prime p and the exponent k.

Fix p :

I(p^k) = g(c_1, k) = I(c^k) where {c_1} = a fixed Euler prime p

Fix k :

I(p^k) = g(p, c_2) = I(p^{c_2}) where {c_2} = a fixed exponent k of the Euler prime p

Note that:

I(p^k) is a monotone decreasing function of p (for constant k)

I(p^k) is a monotone increasing function of k (for constant p)

In both cases, f(p^k) = I(p^k) is a monotone rational function of the parameters p and k.

It follows that, for both cases, f is injective. (That is, "counting" will be possible.)

In order for f to be bijective, f has to be surjective.
(That is, in order to guarantee existence of the pre-image(s).)

Given a rational number a/b, does the equation I(x) = a/b always have (at least) one solution?

To this end, we have the following definitions of terms:

Definition A.0 Let a rational number q = a/b be given in lowest terms (i.e. gcd(a, b) = 1). Let x be in Z+.

(i) If I(x) = q has no solution, then q is called an abundancy outlaw.

(ii) If I(x) = q has at least one solution, then q is called an abundancy index.

Definition A.1 Let a rational number q = a/b be given in lowest terms (i.e. gcd(a, b) = 1). Let x be in Z+.
Suppose q is an abundancy index.

(i) If I(x) = q has exactly one solution, then x is called a solitary number.

(ii) If x is one of at least two solutions of I(x) = q, then x is called a friendly number.