Mersenne, Fibonacci... are there other cases in which the existence of a given prime implies the existence of another prime? Thinking about why it seems impossible to find a case in which the existence of a given prime implies the existence of another bigger prime, I tried to make a list of cases in which knowing that a number is prime then automatically we know that other one is prime. So far I just can recall two well known non-trivial situations:


*

*Mersenne numbers, if $p=2^n-1$ is prime then $n$ is prime as well. The opposite is not true.


*Fibonacci numbers, if $p=F_n$ is prime then $n$ is prime as well. The opposite is not true.

In both cases $n$ is smaller than $p$.
I would like to ask the following questions:


*

*Did I forget more cases?


*It seems that the existence of the bigger prime $p$ implies the existence of the smaller prime $n$ but I can not recall any cases in which a single (only one required) smaller prime $n$ implies that a bigger prime $p$ automatically exists. Are there any papers regarding that impossibility? Are there such cases?
Thank you!

Update 16/02/2016: as kindly explained in the comments by @DanielFischer and answer by @TitoPiezasIII, there are generalizations of both the Mersenne and Fibonacci numbers that are also part of the list.

There seems to be infinitely more such sequences. 

I. Particular

Starting with $n=0$, note that for the Fibonacci numbers,
$$F_n = 0,1,1,2,\color{red}3,5,8,13,\dots\tag1$$
and $F_4=3$ does not have a prime index. Similarly for the Jacobsthal numbers,
$$J_n = 0, 1, 1, 3, \color{red}5, 11, 21, 43,\dots\tag2$$
with the similar exception $J_4=5$. (Indices for prime $J_n$ is A107036.). We also have the Pell numbers,
$$P_n = 0, 1, 2, 5, 12, 29, 70, 169,\dots\tag3$$
and if $P_n$ is prime, then $n$ is prime.

II. General

If we define the first family of the Lucas sequence, 
$$U_n(P,Q) \equiv \frac{a^n-b^n}{a-b}$$
where $a,b$ are the roots of $x^2-Px+Q=0$, then,
$$U_n(1,-1) = \text{Fibonacci}\\
U_n(1,-2) = \text{Jacobsthal}\\
U_n(1,-3) = \text{A006130}\\
\vdots\\
U_n(2,-1) = \text{Pell}\\
U_n(3,-1) = \text{A006190}\\\vdots$$
so presumably appropriate members of the family behave similarly.
