In the realm of number theory, functions that operate on integers and their divisors often present intriguing challenges. This article delves into one such function, denoted as g(n), which is defined for positive integers n ≥ 2. The function g(n) is defined as one more than the largest proper divisor of n. Our primary goal is to determine the number of integers n within the range 2 ≤ n ≤ 100 for which the iterated function g(g(n)) equals 2. This problem combines the concepts of divisors, function iteration, and number theory, requiring a systematic approach to identify the solutions.
To understand the problem, we first need to dissect the function g(n). Recall that a proper divisor of a number n is any divisor of n excluding n itself. The function g(n) takes an integer n as input, finds its largest proper divisor, and returns one more than that divisor. For instance, consider n = 35. The proper divisors of 35 are 1, 5, and 7. The largest among these is 7, so g(35) = 7 + 1 = 8. This definition lays the groundwork for our exploration. The function g(n) essentially provides a way to map an integer to another based on its divisor structure. This mapping becomes particularly interesting when we consider iterating the function, as in g(g(n)). Understanding the behavior of g(n) for different types of numbers – primes, composites, powers of primes – is crucial for solving the problem. For prime numbers, the largest proper divisor is always 1, making g(n) = 2. For composite numbers, the behavior is more varied and depends on the specific divisors. The iterated function g(g(n)) introduces another layer of complexity, as we need to understand how the output of g(n) itself transforms under the same function. This iterative process can lead to interesting patterns and requires careful analysis to solve equations like g(g(n)) = 2. By breaking down the definition and considering examples, we can start to unravel the intricacies of g(n) and its role in the given problem.
Our objective is to find the values of n for which g(g(n)) = 2. To satisfy this condition, we need to work backward. Let's first consider the scenario where g(x) = 2 for some integer x. From the definition of g(n), this implies that the largest proper divisor of x must be 1. The only integers that satisfy this condition are prime numbers, because prime numbers have only two divisors: 1 and themselves. Therefore, if g(x) = 2, then x must be a prime number. Now, we replace x with g(n), so we need to find values of n such that g(n) is a prime number. This means that the value of g(n) must belong to the set of prime numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ...}. To determine the specific values of n, we consider each possible prime value for g(n) and work backward using the definition of g(n). If g(n) = p (where p is a prime number), then the largest proper divisor of n must be p - 1. This constraint helps us identify the potential forms of n. For example, if g(n) = 2, the largest proper divisor of n is 1, meaning n is a prime number. If g(n) = 3, the largest proper divisor of n is 2, suggesting n could be a power of 2. This reverse engineering approach is crucial for solving the problem. By understanding the implications of g(n) being a prime number, we can systematically narrow down the possible values of n that satisfy the given condition.
To systematically solve the problem, we need to analyze the possible cases for g(n). As established, g(n) must be a prime number for g(g(n)) to equal 2. Therefore, we will examine each prime number that g(n) can take within the range determined by 2 ≤ n ≤ 100. We know that g(n) is one more than the largest proper divisor of n. Let's denote the largest proper divisor of n as d. Then, g(n) = d + 1. We will consider primes p such that g(n) = p.
Case 1: g(n) = 2
If g(n) = 2, then the largest proper divisor of n is 1. This implies that n is a prime number. Within the range 2 ≤ n ≤ 100, the prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. There are 25 prime numbers in this range.
Case 2: g(n) = 3
If g(n) = 3, then the largest proper divisor of n is 2. This means n can be expressed as 2k where k can only be 1, which makes n = 4. So n can be 2^2 = 4. If n = 4, then the divisors of 4 are 1, 2, and 4. The largest proper divisor is 2, and g(4) = 2 + 1 = 3. Thus, n = 4 is a solution.
Case 3: g(n) = 5
If g(n) = 5, then the largest proper divisor of n is 4. The possible values for n are multiples of 4 where the other divisors are smaller than 4. The multiples of 4 are 4, 8, 12, 16, 20, .... If n = 8, the proper divisors are 1, 2, 4, so g(8) = 4 + 1 = 5. Thus, n = 8 is a solution.
Case 4: g(n) = 7
If g(n) = 7, then the largest proper divisor of n is 6. The possible values for n can be multiples of 6. If n = 12, the proper divisors are 1, 2, 3, 4, 6, and g(12) = 6 + 1 = 7. Thus, n = 12 is a solution.
Case 5: g(n) = 13
If g(n) = 13, then the largest proper divisor of n is 12. We need to find n such that 12 is its largest proper divisor. If n = 24, the proper divisors are 1, 2, 3, 4, 6, 8, 12, and g(24) = 12 + 1 = 13. Thus, n = 24 is a solution.
Case 6: g(n) = 19
If g(n) = 19, then the largest proper divisor of n is 18. If n = 36, the proper divisors are 1, 2, 3, 4, 6, 9, 12, 18, and g(36) = 18 + 1 = 19. Thus, n = 36 is a solution.
Case 7: g(n) = 31
If g(n) = 31, then the largest proper divisor of n is 30. If n = 60, the proper divisors include 30, and g(60) = 30 + 1 = 31. Thus, n = 60 is a solution.
Case 8: g(n) = 37
If g(n) = 37, then the largest proper divisor of n is 36. If n = 72, the proper divisors include 36, and g(72) = 36 + 1 = 37. Thus, n = 72 is a solution.
Case 9: g(n) = 61
If g(n) = 61, then the largest proper divisor of n is 60. If n = 120, this is outside our range of 2 ≤ n ≤ 100, so there is no solution in this case.
Case 10: g(n) = 97
If g(n) = 97, then the largest proper divisor of n is 96. If n = 192, this is outside our range of 2 ≤ n ≤ 100, so there is no solution in this case.
By considering these cases, we have identified the values of n that satisfy the condition g(g(n)) = 2. The systematic approach allows us to cover all possible scenarios within the given range. We must now consolidate these findings and count the total number of solutions.
Now that we have analyzed the different cases, we can enumerate the values of n that satisfy the condition g(g(n)) = 2. We found that the solutions fall into two categories:
- Prime numbers: When g(n) = 2, n is a prime number. There are 25 prime numbers between 2 and 100, as listed earlier.
- Composite numbers: We identified specific composite numbers where the largest proper divisor leads to a prime value for g(n). These numbers are 4, 8, 12, 24, 36, 60, and 72. There are 7 such numbers.
To find the total number of solutions, we simply add the number of primes and the number of composite numbers we found:
Total solutions = Number of primes + Number of composite numbers
Total solutions = 25 + 7 = 32
Therefore, there are 32 values of n in the range 2 ≤ n ≤ 100 for which g(g(n)) = 2. This completes our solution, providing a concrete answer to the problem. The process involved understanding the function g(n), recognizing the condition for g(g(n)) to equal 2, and systematically analyzing different cases to identify all possible values of n. The combination of number theory concepts and careful case analysis was crucial in arriving at the final answer.
In conclusion, we have successfully determined that there are 32 integers n in the range 2 ≤ n ≤ 100 that satisfy the condition g(g(n)) = 2. This problem showcased the interplay between function iteration, divisibility, and prime numbers. The methodical approach of analyzing cases based on the prime values of g(n) allowed us to identify all possible solutions within the given range. Number theory problems often require a blend of theoretical understanding and practical application, and this example effectively demonstrates that principle. By dissecting the function g(n), understanding its behavior for different types of integers, and working backward from the condition g(g(n)) = 2, we were able to unravel the solution. This exploration not only provides a solution to the specific problem but also offers insights into the broader realm of number theory and its fascinating challenges. The problem-solving techniques used here can be applied to other similar problems, emphasizing the importance of a systematic and analytical approach in mathematics. This journey through the properties of g(n) and its iterations highlights the beauty and complexity inherent in the study of integers and their divisors.