What is the equivalent of the factorial function, but for exponentiation?

Asked Apr 05 '22 at 18:45

Active Sep 26 '22 at 16:10

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The factorial function multiplies a given number by each number less than itself until reaching one. Does a function, notation, or literature yet exist regarding the idea of raising a given number to the power of each number one smaller until reaching one? If I call this function f, f(4) would be $4^{3^{2^1}}$, f(5) would be $5^{4^{3^{2^1}}}$, and so on. I know that the factorial function has been generalized to decimals, but generalizing powers to decimals generally runs into all of the problems tetration usually faces.

I don't know if it's important to notice that the order of multiplication is irrelevant with the factorial function (i.e. multiplying by 1 at the end is the same as at the beginning) but would not be irrelevant here (1 to any power is 1).

edited Sep 26 '22 at 16:10

Eric Wofsey

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asked Apr 05 '22 at 18:45

Isaac Philo

See A049384. – lulu Apr 05 '22 at 18:49
1

Such a function would be just an artificial curiosity, of no particular use. – quasi Apr 05 '22 at 18:50
Thank you @lulu as the OEIS link brought me to this Wolfram Mathworld page about this exact idea: https://mathworld.wolfram.com/ExponentialFactorial.html – Isaac Philo Apr 05 '22 at 19:14
@quasi Curiosity is a virtue. – Jair Taylor Sep 26 '22 at 16:56
See Exponential Factorial vs Tetration AND Does domination of exponential-factorial by tetration generalize to higher-order hyperoperations? AND Can any number of factorials overtake the tetration function ${^x}10$? AND Tetrofactorial web page. – Dave L. Renfro Sep 26 '22 at 17:18
Thank you @JairTaylor – Isaac Philo Sep 28 '22 at 13:42
And thank you @DaveL.Renfro – Isaac Philo Sep 28 '22 at 13:42

What is the equivalent of the factorial function, but for exponentiation?

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