Compute the minimum \$a(n)>a(n-1)\$ such that \$a(1)+a(2)+\dots+a(n)\$ is prime (OEIS A051935) Background

Consider the following sequence (A051935 in OEIS):


Start with the term \$2\$.
Find the lowest integer \$n\$ greater than \$2\$ such that \$2+n\$ is prime.
Find the lowest integer \$n'\$ greater than \$n\$ such that \$2 + n + n'\$ is prime etc.


A more formal definition:

$$a_n=\begin{cases}2 &amp; \text{if }n=0 \\ \min\{x\in\Bbb{N}\mid x&gt;a_{n-1} \text{ and }\left(x+\sum_{i=0}^{n-1}a_i\right) \text{ is prime}\} &amp; \text{otherwise}\end{cases}$$

The first few terms of the sequence are (please refer to these as test cases):

2, 3, 6, 8, 10, 12, 18, 20, 22, 26, 30, 34, 36, 42, 44, 46, 50, 52, 60, 66, 72, 74, ...


Task

Your task is to generate this sequence in any of the following ways:


Output its terms indefinitely.
Given \$n\$, output \$a_{n}\$ (\$n^{\text{th}}\$ term, \$0\$ or \$1\$ indexed).
Given \$n\$, output \$\{a_1, a_2, \dots, a_n\}\$ (first \$n\$ terms).


You can compete in any programming language and can take input and provide output through any standard method, while taking note that these loopholes are forbidden by default. This is code-golf, so the shortest submission (in bytes) for every language wins.
Brachylog, 13 bytes

~l.&lt;₁a₀ᵇ+ᵐṗᵐ∧


Try it online!

Output is the list of first n terms of the sequence. 

?~l.&lt;₁a₀ᵇ+ᵐṗᵐ∧    Full code (? at beginning is implicit)

?~l.              Output is a list whose length is the input
    &lt;₁            Output is an increasing list
      a₀ᵇ+ᵐ       And the cumulative sum of the output
           ṗᵐ     Consists only of prime numbers
             ∧    No further constraints on output

Explanation for a₀ᵇ+ᵐ:
a₀ᵇ               Get the list of all prefixes of the list
                  Is returned in increasing order of length
                  For eg. [2, 3, 6, 8] -&gt; [[2], [2, 3], [2, 3, 6], [2, 3, 6, 8]]
   +ᵐ             Sum each inner list  -&gt; [2, 5, 11, 19]