OK so today I’m trying problem 12 – find the first triangular number with over 500 divisors. This is the first Project Euler problem I’ve really struggled to find a solution in a reasonable amount of time.

What’s a triangular number? It is the sequence found by summing all the natural numbers, for example the third number is \(1+2+3=6\). Interestingly, it counts objects arranged as a triangle.

First six triangular numbers

This also has closed form \(T_n=\sum_{i=1}^{n}i=\frac{n(n+1)}{2}\).

I started with a brute force approach – iterate through the triangular numbers and test if the number of divisors is greater than 500. I’m using the “numbers” package’s Sigma function to implement divisor function \(\sigma _x(n)=\sum_{d|n}d^x\) where \(\sigma _0(n)\) gives the total number of factors for a given number. That requires a loop, which is going to dominate for large \(n\), so \(O(n^2)\).

library(numbers)

triangular.number <- function(number) {
    number = (number * (number + 1)) / 2
}

num.factors <- 0
i <- 1

while (num.factors < 500) {
    num.factors <- 
        (Sigma((triangular.number(i)), k = 0))
    print(triangular.number(i))
    i <- i + 1
}

Not terribly efficient but it gets the correct answer. So how can we improve the algorithm? Reducing the number of times we repeat the loop would be a good place to start.

Now, \(a(n)\) and \(b(n+1)\) are co-prime – the only positive integer that divides them both is 1. This has a useful property, that \(lcm(a,b)=ab\). Thing is, I can’t see how to incorporate it…