找到第一个超过 500 个除数的三角形数

Question

我正在尝试解决 Java 中的第 12 个欧拉问题，但我似乎真的无法理解这里的问题。该脚本旨在输出具有 500 多个除数的第一个三角形数，如代码中的注释中所述。正确答案应该是“76576500”，而我的脚本输出的答案是“842161320”——相差很大。有谁知道我哪里出错了？感谢所有帮助，谢谢！

public class Script_012
{
/*
    The sequence of triangle numbers is generated by adding the natural numbers. So the 7th 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?
*/
public static void main (String [] args)
{
    boolean enough_factors = false;
    long num = 1;
    long runner = 1;
    int num_of_factors;
    int highest_factors = 0;
    while (!enough_factors)
    {
        num_of_factors = 0;
        for (int i = 1; i < (int) Math.sqrt(num); i ++)
        {
            if ((num % i) == 0)
            {
                num_of_factors += 1;
            }
        }
        if (num_of_factors > 500)
        {
            enough_factors = true;
            System.out.println(num);
        }
        runner += 1;
        num += runner;
    }
}
}

Answer 1

问题在于您只是添加小于或等于平方根的因子，但问题是讨论所有因子，包括大于平方根的因子。

简单（但缓慢）的解决方案：

将for (int i = 1; i < (int) Math.sqrt(num); i ++) 更改为for (int i = 1; i <= num; i ++)

更好的解决方案： 保持 for 循环的迭代次数相同，但每次加 2，并仅考虑平方根为一个因素。 代码：

public static void main (String [] args)
{
    boolean enough_factors = false;
    long num = 1;
    long runner = 1;
    int num_of_factors;
    int highest_factors = 0;
    while (!enough_factors)
    {
        num_of_factors = 0;
        for (int i = 1; i < (int) Math.sqrt(num); i ++)
        {
            if ((num % i) == 0)
            {
                num_of_factors += 2;
            }
        }

        if(num % Math.sqrt(num) == 0) {
            num_of_factors++;
        }

        if (num_of_factors > 500)
        {
            enough_factors = true;
            System.out.println(num);
        }
        runner += 1;
        num += runner;
    }
}