A Perfect Factorial Number

We all know that $6$ is a perfect number: $3 + 2 + 1 = 5 + 1 = 6$ But some of us don't know that $3! = 3 * 2 * 1 = 6 * 1 = 6$ So does the number $6$ have extra properties? I believe so. As :

$3 + 2 + 1 = 3!$

$3 + 2 + 1 = 3 * 2 * 1$

$5 + 1 = 6 * 1$

$6 = 6$

So I would like to rename $6$ 's properties and special name from perfect number to perfect factorial number.

Let me know in the comments if you find another number that has similar properties.

#NumberTheory

Easy Math Editor

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Comments

@Yajat Shamji No such number exists between 1 and 10,000 except 6 I used a python program to check it

Zakir Husain - 1 year ago

@Yajat Shamji There is no such number other than 6 (i.e. 6 is unique) see here

Zakir Husain - 1 year ago

Thanks so much, @Zakir Husain! I appreciate your hard work of putting a discussion and proof in to show this and testing a Python program to check. There is a problem titled A Perfect Factorial Number (Problem 1) - check it out and alleviate my doubts (some people think that this is the definition of a perfect number: $6 = 1 + 2 + 3 + 6 = 12$ )

A Former Brilliant Member - 1 year ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$