22257_JohnnyHatesNumberTheory

Johnny hates Number Theory! Actually, back in 2002, we came to know that Johnny couldn’t count and in 2005 we knew that Johnny couldn’t yet add. (But we did know in 2003 that Johnny was street smart enough to solve difficult graph problems!) Why Johnny decided to study Number Theory is incomprehensible to us.
Anyhow, back to Johnny. Johnny just failed his comprehensive exam and that was all because of Euler’s Totient function (ϕ). Johnny is so angry that he decides to create his own Totient function. Here’s how he described it to his advisor:
In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. Johnny defines function F(n), for n ≥ 2, to be the non-decreasing list of prime numbers whose product is n. For example, F(8) = <<2, 2, 2>>, F(60) = <<2, 2, 3, 5>>, and F(71) = <<71>>  (71 is a prime.) Let O(n) be the length of the list F(n) (i.e. its ordinal.) For example, O(8) = 3, O(60) = 4, and O(71) = 1. Johnny also defines function p(n) over positive integers as follows:

The following table illustrates p(n) for the first twenty positive integers:

Given two positive integers a and b where a ≤ b, Johnny defines his very own Totient function ϕ (a, b) as follows:

For example, ϕ(1, 4) = −4, ϕ(16, 16) = 3, and ϕ(8, 12) = 4.
For his dissertation, Johnny needs a program that determines the maximal ϕ within a given range [L,U]. In other words, given two positive integers L,U such that L ≤ U, the program must find the maximum ϕ(a, b) where L ≤ a ≤ b ≤ U. For example, the maximal ϕ within the range [1, 20] is 7 (which is ϕ(8, 16).)
Write the program Johnny needs!