Number of positive divisors of n

**alexmahone** · August 17th 2011, 09:48 AM

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

**Plato** · August 17th 2011, 10:05 AM

Originally Posted by alexmahone

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 $n=144~?$

**alexmahone** · August 17th 2011, 10:35 AM

Originally Posted by Plato

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.

**TheChaz** · August 17th 2011, 10:56 AM

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

**alexmahone** · August 17th 2011, 10:58 AM

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

**Plato** · August 17th 2011, 10:59 AM

Originally Posted by alexmahone

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 $d(144)=(4+1)(3+1)=15$ .

**alexmahone** · August 17th 2011, 11:01 AM

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

**Also sprach Zarathustra** · August 17th 2011, 11:04 AM

Originally Posted by alexmahone

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

Thread: Number of positive divisors of n