# A sequence of sorts

## Main Question or Discussion Point

1
2
1 2
4
1 1 3
6
1 2 4
???

What comes next?

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Moo Of Doom said:
1
2
1 2
4
1 1 3
6
1 2 4
???

What comes next?
8 (a partition of 8, but the even numbers are partitioned into a single partition). I haven't figured out how the odd numbers are partitioned, but it seems that the entry for 2^n - 1 is 1 2 4 ... 2^(n-1)

Nope, that's not it. But that's an interesting observation about the partitions! I hadn't noticed that.

8, i am not sure what the real pattern is but looking from the most left number in each line, 1,2,1,4,1,6,1,... one could easily guess 8

Worth a shot.

It increases the first number by one then multiply it with the rest.
ie:
1 2 4
(1+1) * 2 * 4 = 16

Spoiler warning:

The answer is 16. Or so I think, if my way is allowed.

1
2
1 2
4
1 1 3
6
1 2 4
???

seperate the lines into groups just to understand it better:

1
2
--------
1 2
4
--------
1 1 3
6
------
1 2 4
???

Multiply the top lines digits by themselves and then by two:
(a*b)*2

so the first line:
(1)*2=2

and the second & third:
(1*2)*2=4

(1*1*3)*2=6

So for the bottom line you get:
(1*2*4)*2=16

so you are saying that the top line in each group generated by random ?

Nope, none of the answers so far are correct. Jimmysnyder was the closest with his comment about the partitions.

Maybe there isn't quite enough information.

Here's the pattern a bit longer:

1
2
1 2
4
1 1 3
6
1 2 4
2 6
1 3 5
10
???

What comes next?

Sure. Here's quite a bit more of the sequence:

1
2
1 2
4
1 1 3
6
1 2 4
2 6
1 3 5
10
1 1 1 2 6
12
1 5 7
???

Still want a clue? Partitions are correct. But think of the sequence as starting from 2 rather than 1...

a 16 year olds guess....lol

4 10
1 2 4 8
16

??

Last edited:
Close. The second two are correct, but 4 10 is not. Care to explain your answer?

ummm lol nope - im not good at explain them

ok its sumthin like 4 10...but the numbers add up to 14? i spent about 20 mins workin it out lol ill get it when i get time

Yeah, the numbers add up to 14 but they're not 4 10.

is it 2 12?

and *this is gna seem really stupid* but whats partition?

0m3ga said:
and *this is gna seem really stupid* but whats partition?
A partition is simply a representation of a positive integer as a sum of other positive integers. For instance 12 = 2 + 10 or 12 = 4 + 8, or 12 = 1 + 2 + 9 or in the degenerate case, 12 = 12. The sequence that Moo has presented us with can be viewed as a sequence of partitions. Here is a list of all the possible partitions of 5:

5 = 1 + 1 + 1 + 1 + 1
5 = 1 + 1 + 1 + 2
5 = 1 + 1 + 3
5 = 1 + 4
5 = 1 + 2 + 2
5 = 2 + 3

I hope I didn't miss any. By the way, I didn't know what a partition was until I was about 21, so now you are 5 years ahead of me.

0m3ga said:
is it 2 12?
Nope. First number is 2, though.

the only reasonable guess i can think off is 2 6 6

Now you're just firing blindly :P

Nope, not 2 6 6.

lol yeah howd you guess

i see the whole pattern the only bit i didnt get was the bit where it went 2 6 lol and im guessin thats linked to the line i cant get

Looking at your pattern I think it has something to do with the factorisation of the next number

I can see that each number that is followed by a prime number is left intact

I think it is based on some method of decomposing the fractions

1/2, 2/3, 3/4, 4/5 etc

I notice that 2/3 can't be broken down
but 3/4 = 1/2 + 1/4
7/8 and 15/16 seem to match your breakdown

when I do 8/9=2/3+2/9 it seems to confirm your partition pattern

BUT

9/10=1/2 +4/10 DOESN'T match up

So I'm missing some other criterion that is not apparent

NateTG
Homework Helper
Let's see here:

2 3 10
1 2 4 8
16
1 1 3 3 9
18

NateTG has almost got it. His 2 3 10 doesn't add to 14, though... so that's not quite right. Everything else is correct though.

2 2 10?....