Highly Divisible Triangular Number

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:
$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$
Let us list the factors of the first seven triangle numbers:
\begin{align}
\mathbf 1 &\colon 1\\
\mathbf 3 &\colon 1,3\\
\mathbf 6 &\colon 1,2,3,6\\
\mathbf{10} &\colon 1,2,5,10\\
\mathbf{15} &\colon 1,3,5,15\\
\mathbf{21} &\colon 1,3,7,21\\
\mathbf{28} &\colon 1,2,4,7,14,28
\end{align}
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?

Let’s denote T_n as the n-th triangle number, which is the sum of the first n natural numbers.

So, T_n = 1 + 2 + 3 + … + n = n*(n+1)/2.

Now, a key point is that the number of divisors of two coprime numbers (numbers for which the only common positive factor is 1) m and n is equal to the product of the number of divisors of m and n.

For any triangle number T_n, n and (n+1) are always coprime. So, the number of divisors of T_n is the product of the number divisors of n and (n+1). Therefore, in order to find a triangle number with over 500 divisors, we need to find a pair (n;n+1) such that d(n)*d(n+1) > 500, where d(n) is the number of divisors function.

This simplifies the problem significantly, as now we only need to find the number of divisors of the natural numbers, rather than the triangle numbers.

By calculating the number of divisors for natural numbers, we eventually come to n = 12375, where T_n has 576 divisors. Thus, the first triangle number that has over 500 divisors is T_12375.

To calculate the T_n, we use the formula, T_n = n*(n+1)/2,
which gives us, T_12375 = 12375*12376/2 = 76,576,500.

So, the value of the first triangle number to have over five hundred divisors is 76,576,500.

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