Is this sequence monotonically decreasing?

Question

Let $a_n = \frac{p_n - p_{n-1}}{p_n \log p_n}$ where $p_n$ denotes the $n$-th prime. Is this sequence decreasing (or decreasing after some $N$)?

Answer 1

The sequence is not decreasing.

You have for example:

$$a_4=\frac{2}{7 \cdot \log 7}=0,14 \dots$$ $$a_5=\frac{4}{11 \log 11}=0,15 \dots$$

Doing more examples,you will see that the sequence is getting in general smaller,but it is not monotone..

Here you can see a plot:

EDIT: Consider two twin primes,for example these ones: $3,5$ and $5,7$.

Then it will be like that: $$a_n=\frac{5-3}{5 \cdot \log{5}}=\frac{2}{5 \cdot \log{5}}$$

$$a_{n+1}=\frac{7-5}{7 \cdot \log{7}}=\frac{2}{7 \cdot \log{7}}$$

At this case, $a_{n}>a_{n+1}$.

So,you can't conclude that the sequence is decreasing,because then the relation $\frac{a_{n+1}}{a_n}>1$ would stand $\forall n \in \mathbb{N}$.