# Another Number Sequence

## Main Question or Discussion Point

What is the next number in the following sequence:

4, 6, 12, 27, 60, 138

Up to three yes or no questions about the sequence will be answered.

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What is the next number in the following sequence:

4, 6, 12, 27, 60, 138

Up to three yes or no questions about the sequence will be answered.
Can it be expressed as a purely mathematical function? That is, F(1) = 4, F(2) = 6, etc, where F() is not reliant on non-mathematical expressions such as "the number of vowels in the spelling of some other number", etc.

DaveE

Can it be expressed as a purely mathematical function? That is, F(1) = 4, F(2) = 6, etc, where F() is not reliant on non-mathematical expressions such as "the number of vowels in the spelling of some other number", etc.

I believe not. Although, that does not mean that this string of numbers is something that I randomly came up with. There is a method.

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I believe not. Although, that does not mean that this string of numbers is something that I randomly came up with. There is a method.
Oh, I don't doubt that there's a method or anything-- I just wanted to clarify before I wasted my time checking the number of factors of each element, or how they related to each other mathematically. Actually, truth be told, I probably won't spend any time trying to figure it out beyond now, given that it could really be anything. It could follow months in the year, the number of lights on a digital clock, the order of US Presidents, functions on prime numbers beginning with certain letters, atomic weights on the periodic table, or something equally arbitrary. I'll wait until there's more information-- hopefully someone else will come up with a good question to point in the right direction.

DaveE

Oh, I don't doubt that there's a method or anything-- I just wanted to clarify before I wasted my time checking the number of factors of each element, or how they related to each other mathematically. Actually, truth be told, I probably won't spend any time trying to figure it out beyond now, given that it could really be anything. It could follow months in the year, the number of lights on a digital clock, the order of US Presidents, functions on prime numbers beginning with certain letters, atomic weights on the periodic table, or something equally arbitrary. I'll wait until there's more information-- hopefully someone else will come up with a good question to point in the right direction.

DaveE
Very well, I shall give you a hint: primes.

Is the sequence a transformation of the sequence of the smallest prime numbers, or is it an entirely new sequence where the choice of the next number depends on primality

Is the sequence a transformation of the sequence of the smallest prime numbers, or is it an entirely new sequence where the choice of the next number depends on primality
Not that I know of.

or is it an entirely new sequence where the choice of the next number depends on primality
Yes.

You guys are not fun. I shall even go so far as to make this problem multiple choice.

A) 308
B) 310
C) 312
D) 314

What is the next number in the following sequence:

4, 6, 12, 27, 60, 138

Up to three yes or no questions about the sequence will be answered.
Can the sequence be predicted with one variable i.e. 1,3,5,7,9.... = 2n+1 (n=1 to infinity)?
More than one variable would be something like (2n+1)/(3n2-2).

Can the sequence be predicted with one variable i.e. 1,3,5,7,9.... = 2n+1 (n=1 to infinity)?
More than one variable would be something like (2n+1)/(3n2-2).
Both of your examples are only of 1 variable, but anyway, he already said that the numbers cannot be expressed as a mathematical function, nor a transformation of the series of primes, but rather the numbers are determined based on some primality test.

CRGreathouse
Homework Helper
I think it's 312, but I'm not sure. The 60 throws me off.

I think it's 312, but I'm not sure. The 60 throws me off.

Both of your examples are only of 1 variable, but anyway, he already said that the numbers cannot be expressed as a mathematical function, nor a transformation of the series of primes, but rather the numbers are determined based on some primality test.
Not unless you know of a formula that only generates prime numbers.

Both of your examples are only of 1 variable, but anyway, he already said that the numbers cannot be expressed as a mathematical function, nor a transformation of the series of primes, but rather the numbers are determined based on some primality test.
I knew my question would come out wrong, so that's why I included examples. I meant to ask if more than one variable needed to be used more than once. Although, I already answered my own question.

ƒ(x):

I think it is C - 312 also.

4= 2*2
6= 3*2
12=4*3
27=3*3*3
60=5*4*3
138=23*3*2
312=13*3*2*2*2

The pattern seems to repeat itself. The other choices do not simplify down to these prime numbers.

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60 is not the only exception there. I looked at this too.

One could define some interesting series by using only prime numbers as digits in a multiplicative sort of way...but this isn't what's being done

2*2*2*2
2*2*2*3
2*2*2*5
2*2*2*7
2*2*3*3
2*2*3*5
2*2*3*7
2*3*3*3
2*3*3*5
2*3*3*7
3*3*3*3
3*3*3*5
3*3*3*7
3*3*3*9
5*3*3*3
...

I knew my question would come out wrong, so that's why I included examples. I meant to ask if more than one variable needed to be used more than once. Although, I already answered my own question.

ƒ(x):

I think it is C - 312 also.

4= 2*2
6= 3*2
12=4*3
27=3*3*3
60=5*4*3
138=23*3*2
312=13*3*2*2*2

The pattern seems to repeat itself. The other choices do not simplify down to these prime numbers.
Well, this was not what I had in mind. But, there are many different ways to approach the same problem. I was thinking that saying primes and that the series could not be expressed as a mathematical function unless someone knows a general formula for prime numbers would give someone the idea to look at the actual number sequence and compare it to the sequence of prime numbers.

Between:

4 and 6 --> 1 prime number
6 and 12 --> 2 prime numbers
12 and 27 --> 4 prime numbers
27 and 60 --> 8 prime numbers
60 and 138 --> 16 prime numbers

So...the next number from the listed choices would be 314 (D), since there are 32 primes between 138 and 314.

It seems like it should be:

4, 6, 12, 24, 60, 138
or
4, 6, 12, 28, 60, 138

I threw out proximity to primes right away because 27 wasn't immediately adjacent to any primes.

Admittedly, I suppose other options work too, like:

4, 6, 12, 25, 60, 138
4, 6, 12, 26, 60, 138

Ultimately, that 4th number would make the most sense as 24, 26, or 28 (1 more than the last prime, exactly between the last and next prime, or immediately prior to the next prime). I guess it's kind of disappointing to learn that the sequence is partially arbitrary.

DaveE

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...give someone the idea to look at the actual number sequence and compare it to the sequence of prime numbers.
Well, that was my original thought but I stopped looking at transformations of the sequence of prime numbers from smallest to largest when you answered no to my question,

Is the sequence a transformation of the sequence of the smallest prime numbers?

It seems like it should be:

Ultimately, that 4th number would make the most sense as 24, 26, or 28 (1 more than the last prime, exactly between the last and next prime, or immediately prior to the next prime). I guess it's kind of disappointing to learn that the sequence is partially arbitrary.

DaveE
No need to pout. For all that you know the sequence could have been the number that was four after the prime, unless there was not, in which case it would have been the a countdown until a number was reached. Each of the given numbers, except for 27, directly follow the prime number. Besides, making it multiple choice eliminated any arbitrary conditions that may have existed. 27 was deliberately added instead of 24 to help change this from a textbook problem to one that required more intuition.

CRGreathouse
Homework Helper
2s and 3s in the decimal expansions of the prime factorizations of the numbers.

No need to pout.
I guess what annoys me is that people seem to be interested in stumping others rather than making their problems solvable with the clues given. Anyone can come up with an indecipherable sequence, as has been demonstrated time and time again on this forum. The challenge is making something that's not dead-easy BUT is solvable for your audience. If nobody's getting it, your problem was too difficult, and should give more hints. In the case of sequences, you can almost always provide more numbers.

For all that you know the sequence could have been the number that was four after the prime, unless there was not, in which case it would have been the a countdown until a number was reached.
Except that there weren't enough numbers in the sequence to determine the logic behind choosing 27. The 4th number could have been 24, 25, 26, 27, or 28, and you chose 27 for some reason, but you don't have a mathematical reason, it seems, just arbitrarity. The next time you have a choice is at the 7th number, which can be 314, 315, or 316. And for some magical reason, this time, you went with a number immediately following a prime again, but I don't have enough information to determine why you chose 27.

I understand wanting to make it difficult (though I disagree with the principle), but wouldn't the more legitimate choice be 26 for the 4th term, and 315 for the 7th term? You can guarantee that there will always be a "middle" number between sequential primes, but there won't always be 5 values. If you wanted to go with 27 as the 4th term, you need enough demonstrations of your method of choice (when the next number needs to be decided) for people to deduce the logic. Going to the 8th term would provide a choice of 5 possible terms again, although showing your choosing logic only 3 times is effectively as bad as asking for the next solution to a 3-number sequence, unless it's obvious somehow.

DaveE