# 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

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 $$= 1 + 2 + 9 + 4 + 8 + 1 + 2 + 0 + 1 + 6 + 2 = 36$$

$$a_7 = 252 = 7 \times 36$$

• 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.