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The first few exponential factorials are 1, 2, 9, 262144, ... The number of digits in the exponential factorial of 6 is approximately 5 × 10 183 230.
The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast, [18] in the first work on Faà di Bruno's formula, [19] but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial.
The exponential function is the sum of a power series: [2] [3] = + +! +! + = =!, where ! is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every x {\displaystyle x} per the ratio test .
A googol is the large number 10 100 or ten to the power of one ... (factorial of 70). Using an integral, binary numeral system, one would need 333 bits ...
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
An exponential factorial is an operation recursively defined as =, = . For example, a 4 = 4 3 2 1 {\displaystyle \ a_{4}=4^{3^{2^{1}}}\ } where the exponents are evaluated from the top down. The sum of the reciprocals of the exponential factorials from 1 onward is approximately 1.6111 and is transcendental.
A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k).
In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.