NUMBER OF NUMBERS

Amicable Numbers --- There are a few pair of numbers that have a very peculiar affinity for each other and are so-called "amicable numbers." Take for instance the pair of numbers 220 and 284. It turns out that all the factors of 220, that is those less than itself, add up to 284. And, surprisingly, the factors of 284 add up to 220. I only know of three other pairs like these: 1,184 and 1,210 (discovered by a 16-year-old Italian named Nicolo Paganini), 17,296 and 18,416, and the large pair 9,363,584 and 9,437,056. Can you find others?

1. Perfect numbers :
   A number which is equal to the sum of all its divisors smaller than the number itself is called a perfect number. 
   The first perfect numbers are 6, 28 and 496. They are triangular numbers like every perfect number. 
   
   
   "Triangular" numbers are those which are generated by 
   
   n(n+1) /2...
   
   
   Eg :    1, 3, 6, 10, 15, 21  …
   
   Now   π   -2  =  1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - ...
   
   
   Figurate Numbers  
   You can generalize the triangular numbers and go further to quadrilateral, pentagons, ... 

   triangular numbers  square numbers  pentagonal numbers hexagonal numbers  heptagonal numbers octagonal numbers ...

   n*(n+1)/2  n² n*(3n-1)/2 n*(4n-2)/2  n*(5n-3)/2 n*(3n-2) ...

   1 3 6 10 15 21 28...  1 4 9 16 25 36 49...   1 5 12 22 35 51 70...  161528 456691...   1 7 18 34 55 81 112...  1 8 21 40 65 96 133...  ...

   Feigenbaum numbers, e.g. 4.669 ... . (These are related to properties of dynamical systems with period-doubling. The ratio of successive differences between period-doubling bifurcation parameters approaches the number 4.669 ... , and it has been discovered in many physical systems before they enter the chaotic regime. It has not been proven to be transcendental, but is generally believed to be.)
   A cyclic number is an -digit integer that, when multiplied by 1, 2, 3, ..., , produces the same digits in a different order.
   Cyclic numbers are generated by the full repeated primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (Sloane's A001913).