# happy number and prime

A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers[1]).

More formally, given a number , define a sequence , , … where is the sum of the squares of the digits of . Then n is happy if and only if there exists i such that . If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of the sequence are unhappy. For example, 19 is happy, as the associated sequence is: 12 + 92 = 82 82 + 22 = 68 62 + 82 = 100 12 + 02 + 02 = 1. The happy numbers below 500 are: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496

calculation:

```
@dead_list = []
@cache = []
def happy? num
result = num.to_s.split('').inject(0){|sum,x| sum + x.to_i ** 2}
return false if @dead_list.include? result
if result == 1
@cache = []
return true
end
if @cache.include? result
@dead_list << result
return false
end
@cache << result
return happy?(result)
end
def prime? num
return false if (num % 2 == 0) or (num % 3 == 0) or (num % 5 == 0)
end_num = Math.sqrt(num).ceil.to_i
(2..end_num).to_a.each{|x| return false if num % (x+1) == 0}
end
```

and we can also embed to String

```
class Fixnum
class << self; attr_accessor :dead_list; end
attr_accessor :cache
load 'happy_prime.rb'
def is_happy?
@cache = []
@dead_list ||= []
happy? self
end
def is_prime?
prime? self
#or
#require 'mathn'
#self.prime?
end
end
```