tahsin's blog

By tahsin, history, 15 months ago, In English,

What is the upper bound of total number of divisors of divisors of a number ?

 
 
 
 
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15 months ago, # |
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If a is divisor of x and y and both x and y are divisors of z, do you count a twice?

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    15 months ago, # ^ |
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    yes

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      15 months ago, # ^ |
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      then you may consider the upper bound to be , as the upper bound on number of divisors is (verified upto n = 1018)

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        15 months ago, # ^ |
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        But where does come from? Do you assume that divisors of n are of magnitude ? That isn't true.

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          15 months ago, # ^ |
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          I don't know if I was high writing that... My bad :(

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        15 months ago, # ^ |
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        Let f(n) be the number of divisors of divisors of n. If we are going to use a bound for d(n), we may use the identity:

        where rad(n) and ω(n) are the product and the number of distinct prime divisors of n, respectively. That formula can be obtained by noting that f is multiplicative (being the Dirichlet convolution of d and 1, or the triple Dirichlet convolution of 1), and multiplying everything after getting .

        Now, using the simple estimate

        (which comes from the fact that

        d(n) = (α1 + 1)·...·(αω(n) + 1)

        and increasing each term by 1 multiplies each bracket by at most 3/2)

        we get

        According to the last column of this table, talking only "competitive programming numbers" into account: this bound is better than the trivial

        bound by ~ an order of magnitude, but should also not be very far from the truth - the worst cases have several prime factors, with only the exponent on the prime number $2$ being significant.

        Of course, the asymptotic behaviour of d(n) has already been well-explained here, and even better on What's New. I couldn't obtain a real-world bound using this kind of approach, though.

        Also,

        Unable to parse markup [type=CF_TEX]

        is buggy here. I think there is a problem with parsing the comments.
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15 months ago, # |
Rev. 3   Vote: I like it +28 Vote: I do not like it

if your purpose is not mathematical , you can approximately find by using brute force (I mean , you don't have a time limit in your computer)