# Number sequence: 1, 2, 9, 48, 120, 162

First time entering a puzzle so I have no idea if what I've made is too easy. But it's not on OEIS at least.

Find the next term!

1, 2, 9, 48, 120, 162, __

• Hmm. Is it purely mathematical or not? Feb 25 '20 at 2:39
• No non-mathematical shenanigans occur in the above sequence. @DonThousand Feb 25 '20 at 2:54
• Actually, I guess it depends what you mean by purely mathematical. I'll probably just drop hints over time if it doesn't get answered. Feb 25 '20 at 3:06
• I mean, can I predict the next term if all I know is math. Honestly, the term that's throwing me off is 120, since it has a divisor of 5, unlike all the other terms. Feb 25 '20 at 3:07
• Then yes, it is purely mathematical. Good luck! Feb 25 '20 at 3:14

## 2 Answers

Strongly inspired by Mahdi Mahmoodian's answer:

$$a_n = n \times s$$ where $$s$$ is the sum of digits in all previous numbers in the sequence

$$a_1 = 1 \rightarrow$$ sum of digits $$= 1$$

$$a_2 = 2 = 2 \times 1 \rightarrow$$ sum of digits $$= 1 + 2 = 3$$

$$a_3 = 9 = 3 \times 3 \rightarrow$$ sum of digits $$= 1 + 2 + 9 = 12$$

$$a_4 = 48 = 4 \times 12 \rightarrow$$ sum of digits $$= 1 + 2 + 9 + 4 + 8 = 24$$

$$a_5 = 120 = 5 \times 24 \rightarrow$$ sum of digits $$= 1 + 2 + 9 + 4 + 8 + 1 + 2 + 0 = 27$$

$$a_6 = 162 = 6 \times 27 \rightarrow$$ sum of digits $$= 1 + 2 + 9 + 4 + 8 + 1 + 2 + 0 + 1 + 6 + 2 = 36$$

The answer is:
$$a_7 = 252 = 7 \times 36$$

• Teamwork! I like it! Feb 25 '20 at 16:08
• Awesome job! You got it. Feb 25 '20 at 22:19
• good job. i didn't think about digits :)) Feb 27 '20 at 18:58

I got an idea but it's not complete yet. I will update it as soon as I find something else:

The $$i$$th number has $$i$$ as the divisor.

$$f(1) = 1 * 1 = 1$$
$$f(2) = 2 * 1 = 2$$
$$f(3) = 3 * 3 = 9$$
$$f(4) = 4* 12 = 48$$
$$f(5) = 5*24 = 120$$
$$f(6) = 6 * 27 = 162$$

Also, I find something that applies to the first 4 number:

The second divisor of $$i$$th number is $$\sum_{n=1}^{i-1}f(n)$$.

Example: $$f(4) = 4 * \sum_{n=1}^{3}f(n) =4 * (1 + 2 + 9) = 48$$

But after $$48$$ it doesn't work.

So after all I know one thing about the answer:

It's divisible by 7.

• Good job for the first to get so close to the mark. Upvote from me. Just missing that last bit of the equation which Michal has finished. Feb 25 '20 at 22:20