# Thread: Number of positive divisors of n

1. ## Number of positive divisors of n

Is it true that for all positive integers n, the inequality $\displaystyle d(n)\leq 1 + log_2 n$ holds?

2. ## Re: Number of positive divisors of n

Originally Posted by alexmahone
Is it true that for all positive integers n, the inequality $\displaystyle d(n)\leq 1 + log_2 n$ holds?
I assume from the title that $\displaystyle d(n)$ is the number of divisors of $\displaystyle n$.
Does that hold for $\displaystyle n=144~?$

3. ## Re: Number of positive divisors of n

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

d(144) = 2 + 1 = 3

$\displaystyle 1 + log_2 144 \approx 1 + 7.17 = 8.17$

It does hold for n = 144.

4. ## 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

5. ## Re: Number of positive divisors of n

Originally Posted by TheChaz
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

6. ## Re: Number of positive divisors of n

Originally Posted by alexmahone
144 = 12^2

d(144) = 2 + 1 = 3

$\displaystyle 1 + log_2 144 \approx 1 + 7.17 = 8.17$

It does hold for n = 144.
Because $\displaystyle 144=2^4\cdot 3^2$ then $\displaystyle d(144)=(4+1)(3+1)=15$.

7. ## Re: Number of positive divisors of n

Perhaps there is a mistake in the book: Is $\displaystyle d(n)\geq 1 + log_2 n$ true?

8. ## Re: Number of positive divisors of n

Originally Posted by alexmahone
Perhaps there is a mistake in the book: Is $\displaystyle d(n)\geq 1 + log_2 n$ true?
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