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Prime connection

Problem 425

Published on Saturday, 27th April 2013, 04:00 pm; Solved by 576

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).