Iterating the tau-function

Abstract. For every natural number greater than 2, the sequence generated by iterating the
tau-function is a strictly monotone decreasing sequence, it stabilizes and at the end reaches 2. The second but last value of the sequence is an odd prime. The question of Imre Kátai is what is the asymptotic distribution of these primes, if any.
Our goal was to analyze every tau-iteration sequence of all natural numbers up to a given bound. We also analyzed the tau-iteration sequence for randomly chosen set of large numbers. For calculating the tau-function, efficient factorization methods are necessary.