Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence?

Question 1

So, $1,1,2,3,5,8,13,21...$ Any connection to primes?…it appears not. However, in between the Fibonacci numbers are how much primes? Let’s see:

$1$ and $1$ : $0$
$1$ and $2$ : $0$
$2$ and $3$ : $0$
$2$ and $3$ : $0$
$5$ and $8$ : $1$
$8$ and $13$ : $1$
$13$ and $21$ : $2$
$21$ and $34$ : $3$
$34$ and $55$ : $5$
$55$ and $89$ : $8$
$89$ and $144$ : $13$

Huh. What could this imply? Let me just close with the same annoying (but wonderful) pattern. $1,2,3,5,8,13,21...$

Question 2

Eyebrow raising indeed, though the pattern does not continue as you suggest. I get
$0, 1, 1, 2, 3, 5, 7, 10, 16, 23, 37, 55, 84, 125, 198$

Remember that the the number of primes has a well known growth rate (https://en.wikipedia.org/wiki/Prime_number_theorem). Since the Fibonacci numbers are relatively spread out, using $n/\log n$ to approximate the number of primes less than $n$ will cause the number of primes between them to behave like the growth rate of the primes.

Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence?

Answer

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