Overview

 

Question

The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be \(1 + 2 + 3 + 4 + 5 + 6 + 7 = 28\). The first ten terms would be:

\[1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...\]

Let us list the factors of the first seven triangle numbers:

1:  1
3:  1,3
6:  1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors. What is the value of the first triangle number to have over five hundred divisors?

click for answer

76576500

Solutions

 

Ruby

 triangleNumbers.rb https://raw.github.com/addamh/euler/master/012/triangleNumbers.rb download
#!/usr/bin/env ruby

require 'prime'

def factors_of(number)
  primes, powers = number.prime_division.transpose
  exponents = powers.map{|i| (0..i).to_a}
  divisors = exponents.shift.product(*exponents).map do |powers|
    primes.zip(powers).map{|prime, power| prime ** power}.inject(:*)
  end
  divisors.sort.map{|div| [div, number / div]}
end

sequence = Enumerator.new do |yielder|
  number = 0
  loop do
    number += 1
    yielder.yield number
  end
end

sequence.each do |i|
  t = i*(i+1)/2
  factors = []
  if t > 1
    factors_of(t).each do |chunk|
     factors.push(chunk[0])
     factors.push(chunk[1])
    end
    if factors.uniq.length > 500
      p t
      break
    end
  end
end
$ time ruby triangleNumbers.rb
real	0m7.742s
user	0m7.715s
sys	0m0.023s