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Another numbers quiz

What is the eleventh number in this sequence?

1, 2, 4, 6, 16, 12, 64, 24, 36, 48

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( 12 comments — Leave a comment )
dougs
Jul. 15th, 2009 07:44 am (UTC)
Compare:
1, 2, 4, 6, 16, 12, 64, 24, 36, 48
1, 2, 4, 8, 16, 32, 64, 128, ...

So what's with the 6, 12, 24 that don't fit?

Looks like the next number might be either 2n-1 or some variant of 2x.3x ... so my guesses are 1024 or 60.

I'd rather have a solid reason and a single prediction, rather than some fumbling and two vague guesses.
dougs
Jul. 15th, 2009 07:45 am (UTC)
D'oh, 60 is not 2x.3y, it's 2.2.3.5 -- it just "fits" the 24,36,48 pattern.
dougs
Jul. 15th, 2009 08:14 am (UTC)
I think I've got it, it might well be 1024 but I have some checking to do.

If I'm right, then 2n-1 is an upper bound for each term, so the next term can't be higher than 2048 and so on. It's one of those "it's the lowest number that does this(n)" things -- it's easy to prove that 2n-1 always does this(n) but (obviously) other numbers can sometimes do it too.

I'm going to have to write a tiny little computer programme to do my checking.

Edited at 2009-07-15 08:14 am (UTC)
dougs
Jul. 15th, 2009 08:42 am (UTC)
It is 1024. The next one isn't 2048 after all -- it's 1035 -- but after that it's 4096.

I don't know how you mark spoilers, but I can describe the property the numbers must have here if you want me to.
dougs
Jul. 15th, 2009 08:42 am (UTC)
Gareth Rees's answer, below, coincides with mine.
dougs
Jul. 15th, 2009 08:58 am (UTC)
Nooooo, I have a bug! I'm only checking for the (n+1)th term amongst numbers higher than the nth term I've found!

So the twelfth item isn't 1035, it's 60. Although the 13th term is 4096, as I said.
nwhyte
Jul. 15th, 2009 10:17 am (UTC)
Yep, that's it.

(NOw, 1035 would fit the series that went 1, 3, 9, 15...)
wwhyte
Jul. 15th, 2009 03:25 pm (UTC)
Is that the list of the second-smallest numbers with only n divisors? If so should it really start with 1?
nwhyte
Jul. 15th, 2009 03:50 pm (UTC)
You aren't thinking oddly enough!
wimble
Jul. 15th, 2009 08:21 am (UTC)
I'm reading it as two interleaved sequences:
1, ..., 4, ..., 16, ..., 64, ..., 36, ..., ? and
..., 2, ..., 6, ..., 12, ..., 24, ..., 48.

The first appears to be squares of 1, 2, 4, 8, 6. But I can't see any reason to select the next number in that sequence, in order to square it.
The second, appears to be simply doubling (after the initial step from 2 to 6).

I'd guess at 100, but it's very much vague handwaving :(
gareth_rees
Jul. 15th, 2009 08:25 am (UTC)
Spoilers: It's the sequence of the smallest numbers with exactly n divisors, so the next number is the smallest with exactly 11 divisors, namely 1024.
nwhyte
Jul. 15th, 2009 08:28 am (UTC)
Yep, that's it! Well done.
( 12 comments — Leave a comment )

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