I am Jacob, the person who originally sent the question and the 46-digit lowest prime p for which tan(p)>p to Matt Parker. Matt mentions me in his video. I just saw your post now, and it is wonderful that you have published your method and results for the next three terms in the series of such numbers. In the discussions that followed Matt Parker's video, somebody did calculate all three of these three extra terms, but I do not know if this was you or someone else.

It is not clear to me whether these are all properly proven primes or probable primes; maybe you can clarify this.

I remember at the time I posed the original question that I estimated that each term could be expected to have about 30 times as many digits as the previous term, which is why the 46000-digit term was surprising, but maybe not statistically impossible.

I suppose there are two questions that might be of interest:
1) Can one prove that the sequence of primes for which tan(p)>p is infinite ? There is some discussion about this on OEIS (A249836)
2) Is there any significance to this sequence? Probably not, but one thought is this: The numerator and denominator of the convergent rational approximations to pi/2 are necessarily mutually prime. The numerators form terms of this sequence so they are primes by definition. Therefore, the statistical spreads of the primes, and the convergents to pi/2 are not fully independent, so it might not be true that the chance of a number being in this sequence can be calculated by multiplying the chance of finding a prime with the chance of finding an unusually good rational approximation to pi/2. It would be interesting to see an analysis of this; something of which I am not capable.

Another thing that it is tempting to ask is whether a 5th term could be found. Matt Parker quotes me in his video as saying that there is no chance of finding a second term, and it did not take long for me to be proven totally wrong about this.

Lastly, since you now have a sequence of 4 integers, it might be nice if you post the series on OEIS. It might be the fastest-growing series in all of OEIS. I do not know how welcoming OEIS would be of a 46000 digit number!