Note: This is an updated-tech version (with one corrected typo) of what used to be this web page, written by Clifford Stern and archived by the Wayback Machine on June 6th, 2011. If the link works for you, please tell me so that I may take down this page. The original archive from the Wayback Machine (with typo included [exercise!]) can be found here.


Let's review the definitions of drivers and guides, as documented in 3630finishes.pdf:
A guide consists of $2^a$ ($a \gt 0$) along with a subset of the prime factors of $\sigma(2^a)$.
A driver is a special type of guide that takes the form $2^a v$ where $v \mid \sigma(2^a)$ and $2^{a-1} \mid \sigma(v)$. The second statement enables 2 to qualify as a driver, and is descriptively termed the downdriver. In addition, 23 · 3 also qualifies as a driver under this definition.

I propose to extend the lexicon of aliquot sequences as follows:
For the odd integer $t \gt 1$, write $\sigma(t) = 2^b \cdot u$ where $u$ is odd. Then $b$ is defined as the 2s count of $t$. ([tooz], not [too-es])
Let $v$ be the product of a subset of the prime factors of $\sigma(2^a)$ and $b$ the 2s count of $v$. Define the class of the guide $2^a v$ to be $a - b$. Here are how various drivers and guides fit into the resulting classes:

Class -1
Even perfect numbers:
2 · 3
2^2 · 7
2^4 · 31
2^6 · 127

Class 0
2^3 · 3 · 5
2^5 · 3 · 7
2^9 · 3 · 11 · 31

Class 1
2^3 · 3

Class 2
2^3 · 5
2^5 · 7

Class 3
2^5 · 3

Let $n_i = 2^a v \cdot s \cdot t$ be a member of an aliquot sequence where $2^a v$ is its guide, $s$ is the product of the odd prime factors of $n_i$ whose exponents are even and $t \gt 1$ the product of the remaining factors. Let a change in the exponent $a$ be termed a mutation. This occurs only when the 2s count of $t$ is equal to or less than the class of $2^a v$. In the former case, the exponent $a$ increases and in the latter, $a$ is reduced to the 2s count of $t$. The stability of a guide depends upon its class: the smaller the class, the more stable the guide. For example, a class 2 guide will mutate if $t$ is the product of two primes of the form $4n+1$ or is a prime of the form $8n+3$ or $4n+1$. But a class 1 guide mutates only when $t$ is a prime of the form $4n+1$. When the class of a driver is zero or -1, a small 2s count of $t$ is not sufficient in itself to effect a change in the exponent $a$ because the 2s count of $t$ is always greater than zero. Help is required from one of the components of $v$ by having its exponent aquire an even power in order to temporarily raise the driver's class above zero. For example, when the 2^2 · 7 driver takes the form 2^2 · 7^2, its class of -1 temporarily increases by 3 (the 2s count of 7) so a mutation will occur when the 2s count of $t$ is 2 or 1.

A sequence decreases in a significant and reliable way only when it is driven by the downdriver. This explains why the overall trend is up: the downdriver is badly outnumbered by those drivers that propel sequences upwards. And to make matters worse, the principle ones are of class zero or -1 and thus more stable than the downdriver, which is of class one. When a long run of the downdriver ends in mutation, the likelyhood is that the next driver that is aquired will be one of the others, sending the sequence upwards. In fact, it is not uncommon for the downdriver to mutate directly to 2^2 · 7. Any of the drivers listed at the top of the main page has the potential to send a sequence beyond the range of practical computation, especially those of class -1. When a sequence is adrift, not propelled by a driver or guide, it is at constant risk of aquiring one that sends it upwards: every time the exponent of 2 changes, there is a chance that the new $2^a$ is accompanied by one or more of the factors of $\sigma(2^a)$, thus producing a driver, or a guide such as 2^3 · 5 or 2^5 · 7. It is inevitable that this will happen as a sequence goes through a long series of mutations.