User blog:GamesFan2000/Factor-Multiple Function (FMF)

The Factor-Multiple Function, or FMF, is a mathematical function devised by myself. This function can create some huge numbers.

Rules
Define FM(n) as a series of expressions based on n and its factors and multiples. The following rules are to be used:

Rule 1: Zero is considered illegal, as well as all of the negative integers. The natural numbers are the only legal numbers for this function.

Rule 2: FM(1) will default to 1.

Rule 3: For FM(n) in which n is larger than 1, take all factors except for 1 and all multiples up to and including the nth multiple that isn’t n, as well as all of the non-shared multiples up to the nth non-shared multiple for any factors not equal to n.

Rule 4: All of the numbers you’ve determined in Rule 3 are to be put into an expression. From left-to-right, put in every number from smallest to largest. The operation will be in Knuth’s up-arrow notation. The number of arrows between each number will be equal to the largest number in the expression. Solve the expression. Call the answer a1.

Rule 5: For a1, using the exact same rules as before, except that a1 is used for determining the numbers in the expression this time, create another expression and solve. Call the answer to this new expression a2. For all n>1, keep on creating and solving these expressions up to and including an-1. Once you solve the expression created by an-1, the function is complete. Call the final answer an.

Examples
FM(2) is the first non-trivial expression of the function. Since 2 is a prime, the only legal factor for the expression is itself. The first 2 multiples of 2 larger than 2 are 4 and 6. Since there are no factors that are not equal to 2, we can build the expression. It comes out to 2^^^^^^4^^^^^^6. Yep, we’ve already created a number that is inexpressible in standard form, and we still have one more expression to build. Using a1, the answer of 2^^^^^^4^^^^^^6, we need to build another expression with the same parameters as the first expression, except that a1 is the number being used to find the factors and multiples instead of 2. The answer of the expression that is eventually created is a2, which is the final answer of FM(2). The number is a finite value, but because of the immense size of the first expression, we can’t even express it properly with up-arrows.

FM(3) is the next expression. 3 is prime, and the first 3 multiples are 6, 9 and 12. As such, a1 is the answer of 3^^^^^^^^^^^^6^^^^^^^^^^^^9^^^^^^^^^^^^12. I don’t think I need to tell you how massive this number is. It speaks for itself. a1 will be used to create the expression that gives us a2, which is then used to give us a3, the final answer.

FM(4) is the first expression with a composite n. The factors in play are 2 and 4, and the multiples in play are 6, 8, 10, 12, 14, 16, 18, and 20. 6, 10, 14, and 18 are multiples of 2 that aren’t shared with 4, which is why they are part of the expression. The expression for a1 is 2^204^206^208^2010^2012^2014^2016^2018^2020. We continue the process until we get a4.

FM(6) is the first expression where n has multiple factors that aren’t equal to n, which are 2 and 3. The numbers in use are 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 30, 33, 36, 39, and 42. At this point, it becomes extremely cumbersome to create the expressions. Likewise, solve for a1, and continue until you find a6.

Growth Rates?
Now, because of the nature of the function and the fact that I can’t compare to the FGH to save my life, I’ll need some help determining the growth of the function. Feel free to give me bounds or proofs for this function.