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Problem 12

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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:

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots $$

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

$$\begin{aligned}
\bf{1:}\ & 1\\
\bf{3:}\ & 1,3\\
\bf{6:}\ & 1,2,3,6 \\
\bf{10:}\ & 1,2,5,10 \\
\bf{15:}\ & 1,3,5,15 \\
\bf{21:}\ & 1,3,7,21 \\
\bf{28:}\ & 1,2,4,7,14,28 \\
\end{aligned}$$

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?

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有很多约数的三角形数

三角形数是通过累加自然数所得到的数。例如，第$7$个三角形数是$1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$。前十个三角形数分别是：

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots $$

列举出前七个三角形数的所有约数：

$$\begin{aligned}
\bf{1:}\ & 1\\
\bf{3:}\ & 1,3\\
\bf{6:}\ & 1,2,3,6 \\
\bf{10:}\ & 1,2,5,10 \\
\bf{15:}\ & 1,3,5,15 \\
\bf{21:}\ & 1,3,7,21 \\
\bf{28:}\ & 1,2,4,7,14,28 \\
\end{aligned}$$

可以看出，$28$是第一个约数数量超过五的三角形数。

第一个约数数量超过五百的三角形数是多少？

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