Curiosity: $\text{antiprime} = \text{prime} + 1$

Question

The following is just a mathematical curiosity that popped into my head that I thought was interesting. I haven't been able to find anything about it online, although maybe I just am unaware of the terminology.

One of my favorite numbers is $23$ since $23$ itself is prime yet $24 = 23+1$ is antiprime (or highly composite although I find the name antiprime more catchy). It then occurred to me that quite a few of the twenty smallest antiprimes are one more than a prime:

$$ \begin{array}{cc} \text{antiprime} & \text{antiprime}-1 \\ \hline 1 & \color{red}0 \\ 2 & \color{red}1 \\ 4 & \color{blue}3 \\ 6 & \color{blue}5 \\ 12 & \color{blue}{11} \\ 24 & \color{blue}{23} \\ 36 & \color{red}{35} & \\ 48 & \color{blue}{47} \\ 60 & \color{blue}{59} \\ 120 & \color{red}{119} \\ 180 & \color{blue}{179} \\ 240 & \color{blue}{239} \\ 360 & \color{blue}{359} \\ 720 & \color{blue}{719} \\ 840 & \color{blue}{839} \\ 1260 & \color{blue}{1259} \\ 1680 & \color{red}{1679} \\ 2520 & \color{red}{2519} \\ 5040 & \color{blue}{5039} \\ 7560 & \color{blue}{7559} \\ \end{array} $$ where blue denotes prime while red denotes not prime. Since many results are known about antiprimes and many more, of course, are known about primes, I'd like to ask the community the following questions.

1. Is this just an anomaly due to only having checked the first few antiprimes?
2. Are there infinitely many antiprimes that are one more than a prime?
3. As a follow up two question 2, does anyone know of an algorithm to generate antiprimes that are one more than a prime? I only ask since results are known about the prime facotrizations of antiprimes, so maybe particular factorizations may be of use.
4. I guess the notion of antiprimes and primes are well-defined in a unique factorization domain that is also an ordered ring. So maybe any results in the usual case of the integers can be generalized to such ordered UFDs?
5. Lastly, is there already a name for such antiprime-prime pairs? If not, what should we call them? :)

Again this is just a fun curiosity that I thought I'd ask the community. I have thought a bit about these questions but I have not been able to come up with any results. If anyone has any other interesting questions about these antiprime-prime pairs or know of interesting results that I haven't asked about, please feel free to drop them below.