What is the upper bound of total number of divisors of divisors of a number ?
№ | Пользователь | Рейтинг |
---|---|---|
1 | tourist | 4009 |
2 | jiangly | 3839 |
3 | Radewoosh | 3646 |
4 | jqdai0815 | 3620 |
4 | Benq | 3620 |
6 | orzdevinwang | 3612 |
7 | Geothermal | 3569 |
7 | cnnfls_csy | 3569 |
9 | ecnerwala | 3494 |
10 | Um_nik | 3396 |
Страны | Города | Организации | Всё → |
№ | Пользователь | Вклад |
---|---|---|
1 | Um_nik | 164 |
2 | maomao90 | 160 |
3 | -is-this-fft- | 159 |
4 | atcoder_official | 158 |
4 | cry | 158 |
4 | awoo | 158 |
7 | adamant | 155 |
8 | nor | 154 |
9 | TheScrasse | 153 |
10 | maroonrk | 152 |
What is the upper bound of total number of divisors of divisors of a number ?
Название |
---|
If a is divisor of x and y and both x and y are divisors of z, do you count a twice?
yes
then you may consider the upper bound to be , as the upper bound on number of divisors is (verified upto n = 1018)
But where does come from? Do you assume that divisors of n are of magnitude ? That isn't true.
I don't know if I was high writing that... My bad :(
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
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.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)