I am sure these are not taught at school!

Wonder where you got them from! ]]>

So, you wanted some ideas :

Amicable numbers - where the sum of one numbers factors is equal to another number, and vice versa.

Sociable groups - this is the same as amicable numbers but with 3, 4, 5 or more! The 'order' of a sociable group is how many number are in the sequence. So, perfect numbers are of order-1, sociable groups of order-2 are amicable numbers. But, no groups of order-3 have been found yet. There are, however, many groups of 4, 5 etc.

Smallest group of order-4

1264460, 1547860, 1727636, 1305184

Of order-5:

12496, 14288, 15472, 14536, 14264

Aliquot sequences - every number has its own Aliquot sequence, and lots of Aliquot sequences are the same - they just start in different places. To find a number's Aliquot sequence first add its proper factors (all positive factors excluding the number itself):

20 has proper factors 1, 2, 4, 5, 10, which add to make 22

22 has proper factors 1, 2, 11 which add to make 14

14 has proper factors 1, 2, 7 which add to make 10

10 has proper factors 1, 2, 5 which add to make 7

7 has proper factors 1 which add to make 1

1 has no proper factors, which add to make 0

0 has no proper factors, which add to make 0 ...

An Aliquot sequence terminates when it reaches a number already found (forming a sociable group) or when 1 is reached, because the next number will automatically be zero.

A perfect number's aliquot sequence consists only of itself.

A prime number's aliquot sequence consists of itself and 1.

An amicable number's aliquot sequence consists of itself and its amicable partner...

There are also Aliquot sequences that go forever (at least, nobody knows when they will stop)!

]]>Advanced is blue!

Maybe you could go on from amicable numbers to have topics on sociable groups and Aliquot sequences?

]]>This is the first page in the "Advanced" series - you will notice new graphics. "Advanced" may not be the best word, but I can't think of a better one, suggestions welcome.

Enjoy! And feel free to comment or correct, as usual.

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