## 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.

#### 1 comment:

Arnie Dris said...

The abundancy indices and outlaws are both dense in (1, oo). (That is, between any two two indices (outlaws) you will be able to find another index (outlaw). Recall a similar statement about rational numbers in general.)

Which brings me to three fundamental questions about abundancy indices and outlaws:

(1) Given the equation I(x) = q = a/b where gcd(a, b) = 1, under what conditions on x, a and b is q an index? Likewise, under what conditions on x, a and b is q an outlaw?
(2) If p and q are indices, does it follow that pq is also an index? How about p + q? How about if p and q are outlaws, what can be said about pq and p + q?
(3) If p is an index, then what can be said about 1/p? Is it an index or an outlaw?