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Math Help - Number of positive divisors of n

  1. #1
    MHF Contributor alexmahone's Avatar
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    Number of positive divisors of n

    Is it true that for all positive integers n, the inequality d(n)\leq 1 + log_2 n holds?
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    Re: Number of positive divisors of n

    Quote Originally Posted by alexmahone View Post
    Is it true that for all positive integers n, the inequality d(n)\leq 1 + log_2 n holds?
    I assume from the title that d(n) is the number of divisors of n.
    Does that hold for n=144~?
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Number of positive divisors of n

    Quote Originally Posted by Plato View Post
    I assume from the title that d(n) is the number of divisors of n.
    Does that hold for n=144~?
    144 = 12^2

    d(144) = 2 + 1 = 3

    1 + log_2 144 \approx 1 + 7.17 = 8.17

    It does hold for n = 144.
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  4. #4
    Super Member TheChaz's Avatar
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    Re: Number of positive divisors of n

    Uh... no!
    1
    2
    3
    4
    6
    8
    9
    12
    16
    ... all divide 144, so the left hand side is larger than the right hand side
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Re: Number of positive divisors of n

    Quote Originally Posted by TheChaz View Post
    Uh... no!
    1
    2
    3
    4
    6
    8
    9
    12
    16
    ... all divide 144, so the left hand side is larger than the right hand side
    Oops ... I should have done: 144 = 2^4 * 3^2

    d(n) = (4 + 1)(2 + 1) = 15
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    Re: Number of positive divisors of n

    Quote Originally Posted by alexmahone View Post
    144 = 12^2

    d(144) = 2 + 1 = 3

    1 + log_2 144 \approx 1 + 7.17 = 8.17

    It does hold for n = 144.
    Because 144=2^4\cdot 3^2 then d(144)=(4+1)(3+1)=15.
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  7. #7
    MHF Contributor alexmahone's Avatar
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    Re: Number of positive divisors of n

    Perhaps there is a mistake in the book: Is d(n)\geq 1 + log_2 n true?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Number of positive divisors of n

    Quote Originally Posted by alexmahone View Post
    Perhaps there is a mistake in the book: Is d(n)\geq 1 + log_2 n true?
    The divisor bound What’s new
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