perplexus.info :: Just Math : Factorial sequence

Factorial sequence (Posted on 2006-03-02)

Factorials exhibit an interesting trait. The minimum value needed for the length of x! to reach nx, where n is a positive integer, forms an interesting sequence. Let len(x!) = int(log(x!))+1 to account for the extra digit.

What is the relationship between len(x!) and x?

What is the smallest number such that the len(x!)>=8x?

How can I approximate when len(x!) first exceeds/equals nx?

Submitted by Justin

Rating: 4.0000 (3 votes)

Solution: (Hide)

Part 1: len(x!) is approximately x*len(x/e)
Part 2: 271828170
Part 3: As n gets bigger, we gain more accuracy in our approximation. Our approximation of when len(x!)>=nx is xlog(x/e), where log(x/e) is approximately n. So x is approximately 10^n*e where nx is the number of digits in x!, or, n is simply the multiplier.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Proof/Explanation	Jer	2006-03-02 12:48:08
computer exploration	Charlie	2006-03-02 11:19:50

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