User blog:WaxPlanck/k function

The k function is a googological function that uses up-arrows to achieve the growth rate of fw(n) where f is fast growing hierarchy and w is omega and it is significantly more powerful than the Ackermann function, but much weaker than the busy beaver function gets because the busy beaver function is known to dominate when n is between 7 and 16 inclusive. Here are the 5 known and very easy to compute values:

First, k(-2): = -1 In this example, we are using the successor function to add 1 to -2, which makes -1.

Next, k(-1) = 0. In this example, we subtract -1 by itself, leaving us with 0.

Next, k(0) = 0. In this example, 0 is being multiplied by itself 0 times, keeping the output 0.

Next, k(1) = 1. In this example, we multiply 1 by itself 1 time (1 to the power of 1), which leaves us at 1.

Although the k function grows far faster than these values later, we can get a TINY glimpse into the fast growing nature of this function. Last, k(2) = 4. We are now at tetration, but we can't get anything but 4 because both inputs are 2, which always reduces to 2 added to 2.

k(3) is a tritri, and k(4) is larger than g1 in Graham's Number.