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Totient Chains

Problem 214

Published on Saturday, 25th October 2008, 02:00 pm; Solved by 3174

Let φ be Euler's totient function, i.e. for a natural number n, φ(n) is the number of k, 1 ≤ kn, for which gcd(k,n) = 1.

By iterating φ, each positive integer generates a decreasing chain of numbers ending in 1.
E.g. if we start with 5 the sequence 5,4,2,1 is generated.
Here is a listing of all chains with length 4:

5,4,2,1
7,6,2,1
8,4,2,1
9,6,2,1
10,4,2,1
12,4,2,1
14,6,2,1
18,6,2,1

Only two of these chains start with a prime, their sum is 12.

What is the sum of all primes less than 40000000 which generate a chain of length 25?