"Interesting" Sequences Well, here's a question i myself made up and  i thought it's interesting if i share it with everyone.
We call a sequence of natural numbers (for example $a$) Interesting if (all three must be true): 
*
*It was a strictly increasing sequence
*It had an  even number of items
*All of the following inequalities were true: $a_1a_{2n}<a_2a_{2n-1}<...<a_na_{n+1}$ (Where $a_{2n}$ is the last number in the sequence)
Well, now the questions:


*
*If we consider an even amount of prime numbers a sequence, are they Interesting? In the other words, is for any natural number $n>1$, $p_1p_{2n}<p_2p_{2n-1}<...<p_np_{n+1} $(Where $p_n $is the $n$'th prime number.)
*Is there any real number $k$ so that if for any number in the sequence (for example $a_i$), we had $ka_i>a_{i+1}$, then we could assure that our sequence is Interesting?

Now my own works towards the questions:
For the first question i haven't really done anything, except giving some examples which all came true.
For the second question, i've given a counterexample for $k=2$; Consider the sequence: $2, 3, 5, 9, 16, 23$
Here we have $2a_i>a_{i+1}$ but $2*23=46<3*16=48>5*9=45$ which is probably not the thing we want. This also shows that if such number ($k$) exists, it should be smaller than $2$.
We also know that $k>1$ because if it was not, the sequence would not be strictly increasing.
So if $k$ exists, $2>k>1$.

The answer to your first question is no. Consider the set of primes $\{2,3,...,43\}$. Then we have $11*29=319$ and $13*23=299$.
