Natural number n w/ over 2017 divisors, w/ sqrt[n] <= d < 1,01 sqrt[n]

VanBuren

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Prove that there exists a natural number n, which has over 2017 divisors d satisfying:

nd<1,01n\displaystyle √n ≤ d < 1,01 √n

To be honest, I don't really know where to start, again, but I assume that:

n=2a+3b+5c\displaystyle n = 2^a + 3^b + 5^c , where a+b+c>2016\displaystyle a+b+c>2016 (because 1 is also a divisor)
 
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Prove that there exists a natural number n, which has over 2017 divisors d satisfying:

nd<1,01n\displaystyle √n ≤ d < 1,01 √n

To be honest, I don't really know where to start, again, but I assume that:

n=2a+3b+5c\displaystyle n = 2^a + 3^b + 5^c , where a+b+c>2016\displaystyle a+b+c>2016 (because 1 is also a divisor)
Let pk be the kth prime and suppose
u = p12p22p32...pm2=Πk=1k=m   pk2\displaystyle p_1^2\, *\, p_2^2\, *\, p_3^2\, ...*\, p_m^2\, =\, \Pi_{k=1}^{k=m}\,\,\, p_k^2
and
s = u\displaystyle \sqrt{u}
How many prime divisors would s have? How many prime divisors would 1.01 s have? How does this tell you how to pick an n?
 
Let pk be the kth prime and suppose
u = p12p22p32...pm2=Πk=1k=m   pk2\displaystyle p_1^2\, *\, p_2^2\, *\, p_3^2\, ...*\, p_m^2\, =\, \Pi_{k=1}^{k=m}\,\,\, p_k^2
and
s = u\displaystyle \sqrt{u}
How many prime divisors would s have? How many prime divisors would 1.01 s have? How does this tell you how to pick an n?

Well, I think s has 2m\displaystyle 2^m divisors, but 1.01s\displaystyle 1.01*s ?
N would be Πk=1k=m   pk2\displaystyle \Pi_{k=1}^{k=m}\,\,\, p_k^2 , where m>2017\displaystyle m>2017 ?
 
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Well, I think s has 2m\displaystyle 2^m divisors, but 1.01s\displaystyle 1.01*s ?
N would be Πk=1k=m   pk2\displaystyle \Pi_{k=1}^{k=m}\,\,\, p_k^2 , where m>2017\displaystyle m>2017 ?

Sorry, what I should have said is how many divisors does n have, not how many prime divisors. Also, not that p1, p2, p3, ... are the 1st, 2nd, 3rd, ... primes but rather a set of primes such that p1 < p2 < p3 < ...

Now, more ramblings [I'm not sure we are going any where useful]. Let m=0 [i.e. n=1]. The number of divisors is 1=30.
[1]

Let m=1 [i.e. n = p12]. The number of divisors is 3 = 31
[1*[1], p1*[1], p12*[1]] = [1, p1, p12]

Let m=2. The number of divisors is 9=32
[1*[1, p1, p12], p2*[1, p1, p12], p22*[1, p1, p12]]
= [1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22]

Let m=3 The number of divisors is 27=33
[1*[1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22], p3*[1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22], p32*[1, p1, p12, p2, p1 p2, p12 p2, p22, p1 p22, p12 p22]]
=[...]

Now generalize that, i.e.
s=n=Πk=1mpkq\displaystyle s=\sqrt{n} = \Pi_{k=1}^m\, p_k^q
The number of divisors of s is qm. If we let q=pk, then obviously d is greater than or equal to s. Can we find an m so that pkm< 1.01 s. What about m < log(1.01 s)/log(pk)

Is that enough to play around with for a while?

BTW: There should be some easier way to do this. Anybody?


EDIT: Sorry, that should have been (q+1)k I think
 
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Prove that there exists a natural number n, which has over 2017 divisors d satisfying:
nd<1,01n\displaystyle √n ≤ d < 1,01 √n
To be honest, I don't really know where to start, again, but I assume that:
n=2a+3b+5c\displaystyle n = 2^a + 3^b + 5^c , where a+b+c>2016\displaystyle a+b+c>2016 (because 1 is also a divisor)
I admit that I do not understand your notation. But this may help you.
24527611=517655600=N\displaystyle 2^4*5^2*7^6*11=517655600=N has (5)(3)(7)(2)=210\displaystyle (5)(3)(7)(2)=210 divisors.

Every divisor of N\displaystyle N have the form 2a5b7c11d\displaystyle 2^a\cdot 5^b\cdot 7^c \cdot 11^d where 0a4 , 0b2 , 0c6 , 0d1\displaystyle 0\le a\le 4~,~0\le b\le 2~,~ 0\le c\le 6~,~0\le d\le 1

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