I was recently looking at a elementary number theory book and there was a conjecture which amazed me. That conjecture is nothing but Goldbach's conjecture which says that
Every even number greater than $3$ can be expressed as sum of two primes.
Even though it is very simpler to understand, it is very difficult to decipher it.
Here is a stronger version of this guessed by me with the aid of limited computational power,
For every number $n$ greater than $3$ there exist a $k>0$ such that $n-k$ and $n+k$ are prime numbers.
Obviously this is stronger than Goldbach's conjecture since $2n = n - k + n + k$
We define a function $\psi(n)$ which counts number of such pairs for a given number $n$.
Our conjecture is equivalent to saying that $\psi(n)$ is atleast $1$ for $n > 3$. But using computational softwares much more can be conjectured about $\psi(n)$. I attached a plot of $\psi(n)$ till $n=5000$ 
Another way of expressing this is : For each $n>3$,there are infinitely many semi-primes whose prime factors' average is $n$.
It always seems like $\psi(n)$ is bounded below by some positive nonzero function. If we are to approximate this function by a continous function , that function should be highly fluctuating back and forth. For example if we consider vertical strip with narrow width in the graph, it will contain value of $\psi(n)$ and another value $\psi(n)$ which is almost double or triple than former value.
I know that making conjecture without any theoretical basis is not very reasonable even though it provides us a goal to look up and achieve something.
I just wanted know your reactions and insights about this.