Number of Prime Divisors ofϕk(n), whereϕkIs thek-fold Iterate ofϕ

Prime Chains and Pratt Trees

“…Important quantities of study are N (x), the number of prime chains with p k x (k variable), N (x; p), the number of prime chains with p 1 = p and p k /p 1 x, f (p), the number of prime chains with p k = p, and H(p), the length of the longest prime chain with p k = p. Estimates for these quantities have arisen in investigations of iterates of Euler's totient function φ(n) and Carmichael's function λ(n) (e.g. [5], [6], [7], [19], [28], [29], [30]), the value distribution of λ(n) [21], common values of φ(n) and the sum-of-divisors function σ(n) [22], and the complexity of primality certificates ( [8], [33]).…”

“…Pomerance [32] gave another method for producing primality certificates, but it is an open problem whether the Pratt certificate has longer complexity for most primes (see §1 of [32]). Two important statistics of the Pratt tree are the total number of nodes f (p) and the height H(p), the latter being the length of the longest prime chain ending at p. It is known (see [7], [18], [20]) that the number of primes at a fixed level n in the Pratt tree for most p is ∼ (log 2 p) n /n!. The idea is that for most primes p, p − 1 has a multiplicative structure similar to that of a typical integer of its size; namely, p − 1 has about log 2 p prime factors, uniformly distributed on a log log-scale (see [25], Ch.…”

Two remarks on iterates of Euler’s totient function

“…In particular, if h ≤ (log x) 1/2 / log log x (say), then the number of corresponding p ≤ x is at most x o (1) . Summing over all h ≤ (log x) 1/2 / log log x, it follows that the number of primes p ≤ x for which ω(F (p)) ≤ (log x) 1/2 / log log x is also x o (1) . This proves Theorem 1.3 (in a slightly stronger form).…”

“…, h is x o(1) . Having chosen these primes, we see that the number of possibilities for the set of prime factors of n is bounded by 2 h = x o (1) , since the prime factors of n form a subset of the primes dividing F (n). Finally, given the set S of prime factors of n, the number of possibilities for n itself is bounded by Ψ(x, p k ), where k is the size of S .…”

“…Piecing everything together, the corresponding number of possibilities for n is x o (1) . Summing over the x o(1) possibilities for h completes the proof.…”