Problem 425
Prime connection
Two positive numbers A and B are said to be connected (denoted by “A ↔ B”) if one of these conditions holds:
(1) A and B have the same length and differ in exactly one digit; for example, 123 ↔ 173.
(2) Adding one digit to the left of A (or B) makes B (or A); for example, 23 ↔ 223 and 123 ↔ 23.
We call a prime P a 2’s relative if there exists a chain of connected primes between 2 and P and no prime in the chain exceeds P.
For example, 127 is a 2’s relative. One of the possible chains is shown below:
2 ↔ 3 ↔ 13 ↔ 113 ↔ 103 ↔ 107 ↔ 127
However, 11 and 103 are not 2’s relatives.
Let F(N) be the sum of the primes ≤ N which are not 2’s relatives.
We can verify that F(103) = 431 and F(104) = 78728.
Find F(107).
素数连接
两个正整数A和B被称为相连的（用”A ↔ B”表示）如果以下条件之一成立：
（1）A和B的位数相同，且只有一位数字不同，例如123 ↔ 173。
（2）在A（或B）的左侧加一位数字即得到B（或A），例如23 ↔ 223和123 ↔ 23。
我们称素数P是2的亲戚，如果能用相连的素数组成的链连接2和P，且这些素数都不超过P。
例如，127是2的亲戚。其中一种素数串如下所示：
2 ↔ 3 ↔ 13 ↔ 113 ↔ 103 ↔ 107 ↔ 127
然而，11和103不是2的亲戚。
记F(N)是所有≤ N且不是2的亲戚的素数和。
可以验证F(103) = 431以及F(104) = 78728。
求F(107)。