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Another Number Sequence

  1. May 3, 2009 #1
    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.
    Last edited: May 3, 2009
  2. jcsd
  3. May 3, 2009 #2
    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.

  4. May 3, 2009 #3

    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.
    Last edited: May 3, 2009
  5. May 4, 2009 #4
    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.

  6. May 4, 2009 #5
    Very well, I shall give you a hint: primes.
  7. May 4, 2009 #6
    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
  8. May 4, 2009 #7
    Not that I know of.

  9. May 6, 2009 #8
    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
  10. May 6, 2009 #9
    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).
  11. May 6, 2009 #10
    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.
  12. May 6, 2009 #11


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    I think it's 312, but I'm not sure. The 60 throws me off.
  13. May 7, 2009 #12
    Do you mind if I ask what your reasoning is?
  14. May 7, 2009 #13
    Not unless you know of a formula that only generates prime numbers.
  15. May 7, 2009 #14
    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.


    I think it is C - 312 also.

    4= 2*2
    6= 3*2

    The pattern seems to repeat itself. The other choices do not simplify down to these prime numbers.
    Last edited: May 7, 2009
  16. May 7, 2009 #15
    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

  17. May 7, 2009 #16
    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.


    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.
  18. May 7, 2009 #17
    It seems like it should be:

    4, 6, 12, 24, 60, 138
    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.

    Last edited: May 7, 2009
  19. May 7, 2009 #18
    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,

  20. May 8, 2009 #19
    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.
  21. May 9, 2009 #20


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