# The sequence of triangle numbers is generated by adding the natural numbers. So the 7^(th) 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?
#
# ----
#
# k = n(n+1)/2
# p != 2 && p|n   =>  p|k
# p != 2 && p|n+1 =>  p|k
#
# p|k => p|n v p|n+1 
#
# ex: k=1+2+3+4+5+6+7 = 7x8/2 = 7*2^2 => 1, 2,2^2,7,2*7, 2*2*7
# all possible combinations
#
# 0, {2}, {7}, {2,2}, {2,7}, {2,2,7}
#
# see : https://secure.wikimedia.org/wikipedia/en/wiki/Multiset
# for and explanation of multiset coefficients
# 
# Méthode: 1 trouver la decomposition en nombre premiers de n puis de n+1
# enlever une puissance de deux à celui des deux qui est pair
#
# puis le nombre d'élément est une combinaison
#
#

number=500
nbprimes=200000

class PrimeGenerator
    attr_accessor :limit
    attr_accessor :primes

    def initialize(limit=100000)
        @limit=limit
        self.generate
    end

    def generate
        crosslimit=Math.sqrt(limit);
        sieve=[0]*limit;
        i=0;
        while i<limit
            sieve[i]=false
            i+=1
        end
        i=4
        while i<limit
            sieve[i]=1;
            i+=2
        end
        i=3
        while i<crosslimit
            if !sieve[i]
                m=i*i
                while m<limit
                    sieve[m]=true;
                    m+=2*i
                end
            end
            i+=2
        end
        @primes=[]
        i=2;
        while i<limit
            if not sieve[i]
                @primes <<= i
            end 
            i+=1
        end
    end
end

puts "Generate primes"
$primes=PrimeGenerator.new(nbprimes).primes
puts "primes generated"

def decomposition_prime(n)
    res=[0]*$primes.length
    i=0 # index in res
    $primes.each do |p|
        break if p>n

        j=1 # powerfactor
        while ( n % (p**j) == 0 )
            j+=1
        end
        res[i]=j-1

        i+=1
    end
    return res
end

def nb_multi_subset( multiset ) 
    nb=1
    multiset.each do |n|
        nb *= n+1
    end
    return nb
end

def nbdiv(n)
    if n%2 == 0
        x=n/2
        y=n+1
    else
        x=n
        y=(n+1)/2
    end

    res=decomposition_prime(x)
    res2=decomposition_prime(y)

    # puts "=="
    # puts x
    # puts res.to_s
    # puts y
    # puts res2.to_s
    # puts "=="
   
    i=0
    while i<res.length
        res[i] += res2[i]
        i+=1
    end
    return nb_multi_subset( res )
end


n=3637
notfound=true
while notfound
    res=nbdiv( n ) 
    puts %{#{n} (#{n*(n+1)/2}): #{res}}
    if res >= number
        notfound=false
        break
    end
    n+=1
end
puts n*(n+1)/2