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

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!

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

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.