![]() ![]() By adding the function below onto that sum…ĭef func3(n): for each prime p from 2 to seventhRootOf(n): >if(p^7>count++ end def The sum of the result of both functions above gives us the tentative solution, that yields 10 for f(100) and 179 for f(1000). ∃ distinct p,q,r ∈ ℤ+ such that p|N, q|N and r|N AND 1 >for each prime q from nextPrime to n: >for each prime r from nextPrime to n: >if(p*q*r >count++ end defĭef func2(n): for each prime p from 2 to n: >for each prime q from nextPrime to cubedRootof(n): >if(p*q^3>count++ end def If you notice the last number in each of the lists, N, is exactly a product of p times q times r, which are all the unique divisors of N, which means we can rewrite the problem again in three parts as: The problem description also gives us other numbers with exactly eight divisors, so let’s apply our formula to those numbers as well to try and figure out a pattern:ģ0: Ĥ0: Ĥ2: ĥ4: In the example given, 24, the corresponding values for p, q and r are 2, 3, 4 respectively, which is a trivial case. Now that we have a general formula, we need to find values for p, q and r. ![]() Since all the divisors come in unique pairs ( ) that when the two individual components are multiplied together, yield the number N, we can rewrite the formula for the list of eight divisors of N to be. It’s trivial that every number has 1 and itself as divisors, therefore we can reduce the problem to finding the pattern behind. We’re given 24 as an example, with its’ eight divisors being. Let’s start simplifying this problem by temporarily forgoing the “count of numbers” part in “count of numbers not exceeding n with exactly eight divisors.” In order to find the count of numbers in a range with exactly eight divisors, we must first determine if any specific number, N, in that range has exactly eight divisors. there’s no 24 being evenly divided by -3 exactly -6 times). Also, because the problem description only has the positive divisors of positive numbers, we’re restricting our discussion to concern only the positive integers for the rest of this summary (e.g. The problem as initially stated looks to be very difficult, but by using elementary techniques and some of the basic tenets of Number Theory, we can iteratively rephrase the problem until it becomes trivial to solve. Let f(n) be the count of numbers not exceeding n with exactly eight divisors. The ten numbers not exceeding 100 having exactly eight divisors are 24, 30, 40, 42, 54, 56, 66, 70, 78 and 88. ![]()
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