I have been trying to tackle this problem, but I am having difficulty understanding it:
Let φ be Euler’s totient function, i.e. for a natural number n, φ(n) is the number of k, 1 <= k <= n, 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?
My understanding of this is that the φ(5) is 4, 2, 1 – ie the coprimes to 5 are 4, 2 and 1 – but then why isn’t 3 in that list too? And as for 8, I would say that 4 and 2 are not coprime to 8…
I guess I must have misunderstood the question…
Assuming the question is worded badly, and that φ(5) is 4, 3, 2, 1 as a chain of 4. I don’t find any primes that are less than 40m which generate a chain of 25 – I find some chains of 24, but they are relate to non-prime numbers.
‘Iterating the function’ means running the function on it’s own result. Like: φ(5) = 4; φ(4) = 2; φ(2) = 1; Thus, we get your chain of 5-4-2-1. The same with all the other chains.