# Amicable, Perfect, Sociable Numbers and Hidden Relationships in Ecosystems

A little warning for those with an aversion or anxiety when it comes to numbers. The first paragraph is chock full of them, but relax, read it slowly over a cup of tea or a glass of wine, play a little soft music in the background, stoke up your curiosity, ponder a little, and you might find it an alluring launch pad into the second paragraph.

Here goes. There are two numbers less than 1000 that are called amicable numbers- 220 and 284. Amicable numbers are ones that when factored (numbers that divide into it evenly without a remainder), excluding the number itself, add up to the other number. So, 220 is factored into 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Add these up and you get 284. And 284 is factored into 1, 2, 4, 71, and 142 which added equals 220. Pretty cool! It's an interesting relationship between two different numbers. A perfect number is one that has its own factors adding up to itself- like 6, with 1, 2, 3 adding up to 6. Or 28, with 1, 2, 4, 7, and 14 adding up to 28. Extra cool! Sociable numbers are a little more complicated, but not too hard. They form a cycle called an aliquot sequence with a certain period that begins and ends with the same number. An example is 1,264,460. Wow, that's a big number! But here goes- factor 1,264,460 into 1, 2, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340, 3719, 7438, 14876, 18595, 37190, 63223, 74380, 126446, 252892, 316115, 632230, then add these up and get 1547860. Factor this and add them up to get 1727636. Do it again and get 1305184. One more time and you get 1264460! Back to the beginning! Exceptionally cool! The aliquot sequence is 1264460, 1547860, 1727636, 1305184, and 1264460 with a period of 4. This is the only sociable number with a period of 4 or less. There is one with a period of 5, five with a period of 6, and so on. There are only 51 known perfect numbers with 6 being the lowest one. And there are many known amicable numbers with 220 and 284 being the lowest. That's a lot to think about! Many hidden relationships between numbers that go unnoticed and unappreciated by almost all of us.

Just as these numbers have many hidden relationships, so do the many components of an ecosystem, like a bluff prairie or an oak savanna. Only, those ecosystems are way more complicated with the relationships in constant flux as conditions change! Each part of what makes up the whole touches and depends upon other parts in ways that defy precise explanations. The soil is made up of untold numbers of bacteria and fungi which supports the multitude of plants and insects, that provide food and homes for birds, reptiles, amphibians, and mammals. Everything is intertwined in deep relationships that go back many, many millions of years so that even a small disturbance can ripple through the entire ecosystem. Sometimes one part is so important to the whole that it is called a keystone species. If removed, just like if one of the factors of the amicable, perfect or sociable numbers is removed, it can cause a collapse of the whole system. No doubt, nature would eventually solve these problems, but that could take a very long time and leave a host of species in the bin of extinction. The best we can do is to help the ancient patterns of life persist as intact as possible in a world that is in upheaval. So, let's be an amicable, sociable species and reach for perfection to do what we can to save some of the ecosystems like bluff prairies and savannas that need our help. Let's share the earth with all the other creatures that call it home.