Project Euler 387

题目

Harshad Numbers

A Harshad or Niven number is a number that is divisible by the sum of its digits.

$201$ is a Harshad number because it is divisible by $3$ (the sum of its digits.)

When we truncate the last digit from $201$, we get $20$, which is a Harshad number.

When we truncate the last digit from $20$, we get $2$, which is also a Harshad number.

Let's call a Harshad number that, while recursively truncating the last digit, always results in a Harshad number a right truncatable Harshad number.

Also:

$201/3=67$ which is prime.

Let's call a Harshad number that, when divided by the sum of its digits, results in a prime a strong Harshad number.

Now take the number $2011$ which is prime.

When we truncate the last digit from it we get $201$, a strong Harshad number that is also right truncatable.

Let's call such primes strong, right truncatable Harshad primes.

You are given that the sum of the strong, right truncatable Harshad primes less than $10000$ is $90619$.

Find the sum of the strong, right truncatable Harshad primes less than ${10}^{14}$.

解决方案

令 $N=14.$ 不难发现，$1$ 位质数不是可右截强 Harshad 数，因为一位数与自己数位和的商为 $1$，不是一个质数。

本题主要关注可右截 Harshad 数。这种数的有一种生成思想：如果当前数 $x$ 是一个可右截 Harshad 数，并且 $10x+d$（$d$ 是一个新的数位）也是一个 Harshad 数，那么 $10x+d$ 就是一个可右截 Harshad 数。

很明显，$1\sim 9$ 本身就是可右截 Harshad 数，将作为起点进行扩展。最终发现，${10}^{N-1}$ 以内的可右截 Harshad 数的数量不多。

接下来，我们将枚举出来的可右截 Harshad 数，逐一直接判断是否为强 Harshad 数，并保留强 Harshad 数的一部分。

对每个右截强 Harshad 数 $y$，后面都填上 $1,3,7,9$ 这 $4$ 个数位 $d$ 中的一个，那么构造出了 $10y+d$。最终如果 $10y+d$ 是质数，那么就添加到答案中。

代码

from tools import is_prime

N = 14


def is_harshad(n: int):
    return n % sum(int(x) for x in str(n)) == 0


ls = [[1, 2, 3, 4, 5, 6, 7, 8, 9]]
tp = [1, 3, 7, 9]
for i in range(N - 2):
    ls.append([])
    for x in ls[i]:
        for j in range(10):
            t = x * 10 + j
            if is_harshad(t):
                ls[i + 1].append(t)
ans = 0
for v in ls:
    for n in v:
        if not is_prime(n // sum(int(x) for x in str(n))):
            continue
        for y in tp:
            w = n * 10 + y
            if is_prime(w):
                ans += w
print(ans)