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?

+-%echo
echo(Message,Sender)->Sender ! Message.

+-%prime_list
prime_list()->
	case whereis(primeList) of
		undefined->
			Pid=spawn(euler,prime_list,[initialize]),
			register(primeList, Pid),
			Pid;
		Exists->
			Exists
	end.
prime_list(initialize)->
	put(primes,array:from_list([2,3,5,7,11,13,17,19])),
	put(total,8),
	prime_list(idle);
prime_list(idle)->
	Size=get(total),
	receive
		{get,FromZero,Reply} when FromZero < Size ->
			Reply ! {prime,FromZero,array:get(FromZero,get(primes))};
		{get,FromZero,Reply} ->
			echo({get,FromZero,Reply},self()),
			prime_list(gen);
		{all_below,What,Reply} ->
			case array:get(Size-1,get(primes)) < What of 
				true->
					prime_list(gen),
					echo({all_below,What,Reply},self());
				false->
					Reply ! {primes_below,What,prime_list(below,What,0,[])}
			end;
		{is_prime,What,Reply}->
			Skirt=math:sqrt(What),
			Largest=array:get(Size-1,get(primes)),
			case Largest < What of
				true->
					case Largest < Skirt of 
						true->
							prime_list(gen),
							echo({is_prime,What,Reply},self());
						false->
							Reply ! {is_prime,What,prime_list(check_factors,0,What,Skirt)}
					end;
				false->
					Reply ! {is_prime,What,prime_list(binary_search,What,0,Size-1)}
				end
		end,
		prime_list(idle);
prime_list(gen)->
	Size=get(total),
	Start=array:get(Size-1,get(primes))+2,
	End=Start*10+1,
	Sqrt=math:sqrt(End),
	AsList=array:to_list(get(primes)),
	[2|Odds]=AsList,
	RelevantOdds=lists:takewhile(fun(N)->N =< Sqrt end,Odds),
	Newbs=prime_list(gen,Size,Start,End,RelevantOdds,ordsets:from_list(lists:seq(Start,End,2))),
	put(primes,array:from_list(lists:append(AsList,ordsets:to_list(Newbs)))),
	put(total,get(total)+ordsets:size(Newbs)).
prime_list(check_factors,NextIndex,N,Sqrt)->
	NextPrime=array:get(NextIndex,get(primes)),
	case NextPrime > Sqrt of
		true->true;
		false->
			case N rem NextPrime ==0 of
				true->false;
				false->
					prime_list(check_factors,NextIndex+1,N,Sqrt)
			end
		end;
prime_list(binary_search,What,Low,High) when High-Low < 2 ->
	H=array:get(High,get(primes)),
	L=array:get(Low,get(primes)),
	if
		H == What; L == What ->
			true;
		true->false
	end;
prime_list(binary_search,What,Low,High)->
	Mid=Low+((High-Low) div 2),
	Got=array:get(Mid,get(primes)),
	if
		What > Got->prime_list(binary_search,What,Mid+1,High);
		What < Got->prime_list(binary_search,What,Low,Mid-1);
		true->true
	end;

	
prime_list(below,What,Index,SoFar)->
	Next=array:get(Index,get(primes)),
	if
		Next >= What->
			lists:reverse(SoFar);
		true->
			prime_list(below,What,Index+1,[Next|SoFar])
	end.
prime_list(gen,_,_,_,[],Survivors)->Survivors;
prime_list(gen,Total,Lowest,Highest,[MyPrime|RestPrimes],Survivors)->
	prime_list(gen,Total,Lowest,Highest,RestPrimes,
		ordsets:subtract(Survivors,
			ordsets:from_list(
				lists:seq(Lowest-(Lowest rem MyPrime),Highest,MyPrime)))).

+-%prime_iterator
prime_iterator(Pid)->spawn(euler,prime_iterator,[idle,0,2,Pid,prime_list()]).
prime_iterator(idle,Index,Next,Parent,List)->
	receive
		next->
			Parent ! {prime, Next},
			List ! {get, Index+1,self()},
			prime_iterator(wait,Index+1,nothing,Parent,List)
	end;
prime_iterator(wait,Index,_,Parent,List)->
	receive
		{prime,Index,Next}->
			prime_iterator(idle,Index,Next,Parent,List)
	end.

+-%factor
factor(1)->[];
factor(N)->factor(N,prime_iterator(self()),0,math:sqrt(N)).
factor(1,_,_,_)->[];
factor(N,Iter,LastPrime,Sqrt) when LastPrime >= Sqrt->
	exit(Iter,kill),
	[N];
factor(N,Iter,_,Sqrt)->
	Iter ! next,
	Next=receive
		{prime,X}->X
	end,
	if
		N rem Next == 0 ->
			Small=N div Next,
			exit(Iter,kill),
			[Next|factor(Small,prime_iterator(self()),0,math:sqrt(Small))];
		true->
			factor(N,Iter,Next,Sqrt)
	end.

+-%divisor_count
divisor_count(N) when is_integer(N) ->
	divisor_count(factor(N));
divisor_count([])->1;
divisor_count(L)->
	[T|_]=L,
	(length(lists:takewhile(fun(X)->X==T end, L))+1)*divisor_count(lists:dropwhile(fun(X)->X==T end, L)).

+-%triangles
triangles(N)->(N*(N+1))div 2.

p12()->
	p12(1).
p12(N)->
	T=triangles(N),
	C=divisor_count(T),
	if
		C > 500 ->
			io:format("~w~n",[T]);
		true->
			p12(N+1)
	end.

+-#PrimeList
class PrimeList
  @@primes=[2,3,5,7,11,13,17,19]

  def initialize
    @next=-1
  end

  def last
    return 0 if @next==-1
    PrimeList.get(@next)
  end

  def getNext
    PrimeList.get(@next+=1)
  end

  def [](i)
    PrimeList.get(i)
  end

  def PrimeList.isPrime?(n)
    if(@@primes[-1] >= n)
      return true if @@primes.index n
      return false
    end
    p=PrimeList.new
    sq=Math.sqrt(n)
    while(p.last<=sq)
      return false if n % (p.getNext)==0
    end
    true
  end

  def PrimeList.get(i)
    PrimeList.gen while @@primes.length<=i
    @@primes[i]
  end

  def PrimeList.getPrimeNear(n)
    PrimeList.gen until n<@@primes[-1]
    @@primes[PrimeList.primeIndexNear(n)]
  end  

  private

  def PrimeList.primeIndexNear(n)
    return 0 if n<@@primes[0]
    return [-1] if n>@@primes[-1]
    PrimeList.getIndexNear(@@primes,n)
  end

  def PrimeList.getIndexNear(list,n)
    return 0 if list.length==1
    if(list.length==2)
      avg=(list[0]+list[1])/2.0
      return 1 if n>avg
      return 0
    end
    splitAfter=list.size/2
    return PrimeList.getIndexNear(list[0..splitAfter],n) if n<list[splitAfter]
    return splitAfter+1+PrimeList.getIndexNear(list[(splitAfter+1)..-1],n) if n>list[splitAfter+1]
    avg=(list[splitAfter]+list[splitAfter+1])/2
    return splitAfter+1 if n>avg
    splitAfter
  end

  def PrimeList.gen
    add=@@primes[-1]
    cands=Set.new(1..(5*add)){|i|2*i+add}
    theBegin=add+2
    theEnd=11*add
    highest=Math.sqrt(theEnd).to_i
    upTo=PrimeList.primeIndexNear(highest)
    upTo-=1 if @@primes[upTo]>highest
    @@primes[1..upTo].each do |mult|
      ((theBegin-theBegin%mult)..(theEnd-theEnd%mult + mult)).step(mult){|i|cands.delete i}
    end
    @@primes=@@primes+cands.to_a.sort
  end
end

+-#factors
def factors(n)
  return [] if n<2
  p=PrimeList.new
  sq=Math.sqrt(n)
  while(p.last<=sq)
    if n % (p.getNext)==0
      return [p.last]+factors(n/p.last)
    end
  end
  return [n]    
end

+-#divisor_count
def divisor_count(n)
  f=factors n
  mult=1
  while f.length>0
    n=f.pop
    i=2
    while f.length>0 and f.last==n
      i+=1
      f.pop
    end
    mult*=i
  end
  mult
end

def p12
  i=j=1
  i+=(j+=1) while divisor_count(i)<=500
  puts i
end

+-//PrimeList
object PrimeList{
  var myPrimes=Array(2,3,5,7,11,13)
  def get(i: Int)={
    while(i>=myPrimes.length)expand
    myPrimes(i)
  }
  private def expand{
    val base=myPrimes.last+2
    val highest=base+myPrimes.length*10
    val candidates=new Array[Int](1+(highest-base)/2)
    val limit=Math.sqrt(highest)
    val filters=myPrimes.takeWhile(_<=limit).dropWhile(_==2)
    def filterWith(p: Int){
      var x=(base.toFloat/p).ceil.toInt*p
      while(x<=highest){
        if(x%2==1)candidates((x-base)/2)=1
        x+=p
      }
    }
    filters.foreach(filterWith)
    var survivors=List[Int]()
    var i=0
    while(i<candidates.length){
      if(candidates(i)==0)survivors=(base+2*i)::survivors
      i+=1
    }
    myPrimes=myPrimes++survivors.reverse
  }
}

+-//factor
def factor(n: Long):List[Int]={
  if(n==1)return List[Int]()
  val limit=Math.sqrt(n)
  var i=0
  var p=0
  while(p<=limit){
    p=PrimeList.get(i)
    if(n%p==0)return p::factor(n/p)
    i+=1
  }
  List(n.toInt)
}

+-//divisorCount
def divisorCount(n: Long)={
  def process(s: Seq[Int]):Int={
    if(s.length==0) 1
    else (1+s.takeWhile(_==s.first).length)*process(s.dropWhile(_==s.first))
  }
  process(factor(n))
}

def p12{
  var a=1
  var b=1
  while(divisorCount(a)<501){
    b+=1
    a+=b
  }
  println(a)
}