Smith numbers are numbers such that the sum of the decimal digits of the integers that make up that number is the same as the sum of the decimal digits of its prime factors excluding 1.

By definition, all primes are excluded as they (naturally) satisfy this condition!

Smith numbers are also known as   joke   numbers.



Example
Using the number 166

Find the prime factors of 166 which are: 2 x 83

Then, take those two prime factors and sum all their decimal digits: 2 + 8 + 3 which is 13

Then, take the decimal digits of 166 and add their decimal digits: 1 + 6 + 6 which is 13

Therefore, the number 166 is a Smith number.



Task
Write a program to find all Smith numbers below 10000.



See also
 from Wikipedia:   [Smith number].
 from MathWorld:   [Smith number]. 
 from OEIS A6753:   [OEIS sequence A6753].
 from OEIS A104170:   [Number of Smith numbers below 10^n]. 
 from The Prime pages:   [Smith numbers].


require "prime"
 
class Integer
 
  def smith?
    return false if prime?
    digits.sum == prime_division.map{|pr,n| pr.digits.sum * n}.sum
  end
 
end
 
n   = 10_000
res = 1.upto(n).select(&:smith?)
 
puts "#{res.size} smith numbers below #{n}:
#{res.first(5).join(", ")},... #{res.last(5).join(", ")}"