# What is the next number in this sequence: $1, 2, 6, 24, 120$? [closed]

I was playing through No Man's Sky when I ran into a series of numbers and was asked what the next number would be.

$$1, 2, 6, 24, 120$$

This is for a terminal assess code in the game no mans sky. The 3 choices they give are; 720, 620, 180

• What was the purpose of the question? – haqnatural Aug 16 '16 at 17:42
• @Battani I was trying to figure out what the next number in the sequence was. – Atom Aug 16 '16 at 17:43
• @Watson I did when I posted this, I was going to ask this last night but decided to work through it first and ended up solving it. When I saw that neither the question nor answer were on here already I selected the "answer your own question" option when posting the question. That way the question would be available online and I would instead be contributing instead of asking for an answer and providing a hodgepodge of behind the scenes work I was doing. I can delete this if that's not the proper way of doing it! – Atom Aug 16 '16 at 17:58
• oeis.org is a good resource. A search gives several hundred possibilities, but you'd want to go with the most comprehensible. – Teepeemm Aug 16 '16 at 20:30

The next number is $840$. The $n$th term in the sequence is the smallest number with $2^n$ divisors.

Er ... the next number is $6$. The $n$th term is the least factorial multiple of $n$.

No ... wait ... it's $45$. The $n$th term is the greatest fourth-power-free divisor of $n!$.

Hold on ... :)

Probably the answer they're looking for, though, is $6! = 720$. But there are lots of other justifiable answers!

• Woah! Online encyclopedia of integer sequences?! That's amazing. You're right about that. When asked the question originally I was given a few pre-selected answers, I should have taken note and included those to narrow down the options, didn't realize just how many sequences there were! – Atom Aug 16 '16 at 18:00
• It's a gem of a reference. Very, very helpful site! – John Aug 16 '16 at 18:04
• @BernardMasse Having been expected to be able to figure out the next number, there is, logically, a logical method to finding the next number, as trial and error won't get you far, as you can see by the size of $720\dots$ – Simply Beautiful Art Aug 16 '16 at 21:27

After some testing I discovered that these numbers are being multiplied by their corresponding number in the sequence.

For example:

1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120


Which would mean the next number in the sequence would be

120 x 6 = 720


and so on and so forth.

Edit: Thanks to @GEdgar in the comments for helping me make pretty cool discovery about these numbers. The totals are also made up of multiplying each number up to that current count.

For Example:

2! = 2 x 1 = 2
3! = 3 x 2 x 1 = 6
4! = 4 x 3 x 2 x 1 = 24
5! = 5 x 4 x 3 x 2 x 1 = 120
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

• Look up "factorial" on the web to find out more. – GEdgar Aug 16 '16 at 17:42
• Notice what happens if you go downward: You have $720$ and divide by $6$, getting $120$. Then divide by $5$, getting $24$. Then by $4$, getting $6$, then by $3$, getting $2$, then by $2$, getting $1$, then by $1$, getting $1$. Which shows that the factorial of $0$ is $1$. Why the factorial of $0$ is $1$ is something people often find puzzling. $\qquad$ – Michael Hardy Aug 16 '16 at 18:00
• @MichaelHardy Is 0 usually universally considered the factorial of 1? Or is this an exemption where once you get that low that part of the sequence is ignored since it seems odd that 0 is the factorial of 1 and you just start with 1 instead? – Atom Aug 16 '16 at 18:03
• @Atom : $0$ is not the factorial of $1$; rather, $1$ is the factorial of $0$: \begin{align} 0! & = 1 \\ 1! & = 1 \\ 2! & = 2 \\ 3! & = 6 \\ 4! & = 24 \\ 5! & = 120 \\ 6! & = 720 \\ \text{etc.} \end{align} – Michael Hardy Aug 16 '16 at 18:15
• to understand 0!: n! = the nth derivative of x^n. x^0=1, and taking zero derivatives of 1 leaves 1; so 0!=1. – amI Aug 16 '16 at 19:03

The next number is 720.

The sequence is the factorials:

1 2 6 24 120 = 1! 2! 3! 4! 5!

6! = 720.

(Another way to think of it is each term is the term before times the next counting number.

T0 = 1; T1 = T0 * 2 = 2; T2 = T1 * 3 = 6; T3 = T2 * 4 = 24; T4 = T3 * 5 = 120; T5 = T4 * 6 = 720.

• it's yet done. Please find another answer , a little bit original :) perhaps with the sum of the digits ? note also that it begins with 1 2 and ends with 120. Perhaps its an opportunity to concatenate and add zeroes. Good luck – user354674 Aug 16 '16 at 21:28
• Welcome to Math.SE. Glad to have you here contributing. I invite you to check out the MathJax tutorial to make the typesetting for your answers all mathy and stuff. – John Aug 16 '16 at 21:39