Is this Prime Sequence the longest?

So you are interested in Prime Numbers and puzzles thereof. You saw the following on PSE and gave it a try and got it long after the correct answer was posted by @hexomino.

My Eight Cousins

But then you think that you could design another puzzle.

So you start with 2 prime numbers say A and B with B > A

Then you put another Prime C (C>B) in there.

Turns out B is the average of the three numbers! That is nice.

Then you put another Prime D (D>C) with them. You guessed it.

Now C is the average of A,B,C and D!

Then you put another Prime number E (E>D) with them.

Sure enough now D is the average of A,B,C,D and E!

Now you are excited. You think you have discovered something great. Shake up the Math world??

So you continue with another prime F>E. And yes same thing! E is the average of A to F.

You continue these steps until your bubble bursts when you get the number 5917. Oh no. It is not a Prime.

So you wonder. Is this the longest sequence you can do or is there a sequence longer than one you have discovered? How do you explain this anyway? Is there a math equation for this?

So how many total numbers were in this sequence?