Number Theory: Iterated Functions and Prime Factorization Problem

1. Understanding the Definitions

• The function : For any non-zero integer , is defined as the maximum divisor of that is not equal to .
  • If , is the largest proper divisor of .
  • If , let where . The divisors of are the same as the divisors of . The condition means we exclude . Thus, is the largest divisor of smaller than .
  • Note: For any prime , and . For , , so . For , , so .
• The sequence : is the smallest number of iterations of such that the result is .
  • Let denote the -th iteration. is the smallest such that .
  • Example: If (prime), , . So .
  • Example: If , , so .
  • Example: If , , so (as stated in the problem).

2. Analyzing for powers of a prime

• …
• 
  Thus, .

• (since the divisors of are , and we exclude , the max is ).
• …
• 
  Thus, for .
  Special case: .

3. Solving the Problem Statement

1. where . Then .
2. where . Then .

• If , .
• If (with ), .