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Math Help - Sum of divisors function question

  1. #1
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    Thumbs down Sum of divisors function question

    Let p(i) denote the i-th prime number. (Thus (p1, p2, p3,....) = (2, 3, 5, ... ).)
    Find the smallest positive integer k such that the product n = p1 p2 ... pk satisfies
    s(n) > 3n. Is there any positive integer m < n satisfying s(m) > 3m?

    s denotes the sum of divisors function.

    Honestly I have 0 clue on how do I start a question like this...

    It would be nice, if someone can lead me to the right way.

    Thanks
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  2. #2
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    Quote Originally Posted by Khonics89 View Post
    Let p(i) denote the i-th prime number. (Thus (p1, p2, p3,....) = (2, 3, 5, ... ).)
    Find the smallest positive integer k such that the product n = p1 p2 ... pk satisfies
    s(n) > 3n. Is there any positive integer m < n satisfying s(m) > 3m?

    s denotes the sum of divisors function.

    Honestly I have 0 clue on how do I start a question like this...

    It would be nice, if someone can lead me to the right way.

    Thanks
    By the way the sequence of n_i are called primorials.

    I'm afraid I'm getting spoiled by my CAS and got the answer initially just by writing a loop. But reflecting a bit and using reference

    Divisor Function -- from Wolfram MathWorld

    note that s(ab) = s(a)s(b) and that s(p) = p+1 where p prime. So s(p_1*p_2*...*p_i) = (p_1+1)(p_2+1)...(p_i+1). So just build the sequence and stop when the condition is met.

    The answer to the second question is yes, there are many such integers m.
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  3. #3
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    Undefined: Can you explain what the question is asking me?

    I don't get the quesiton at all...
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  4. #4
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    Quote Originally Posted by Khonics89 View Post
    Undefined: Can you explain what the question is asking me?

    I don't get the quesiton at all...
    Sum of divisors of n is what its name suggests.. so s(8) = 1 + 2 + 4 + 8.

    Question asks about comparing

    2 with s(2),
    2*3 with s(2*3),
    2*3*5 with s(2*3*5),
    etc.

    What don't you understand?
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  5. #5
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    why are there many m such that s(m)>3m
    Is it because m is defined as something different to n?
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    Quote Originally Posted by mel240 View Post
    why are there many m such that s(m)>3m
    Is it because m is defined as something different to n?
    For example consider these guys

    Highly composite number - Wikipedia, the free encyclopedia
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