$3n+1$ Collatz generating Chen Primes

Question

I Ask it just for fun:

Consider the Collatz sequence $3n+1$.

$x$ is a Chen-prime-$3x+1$ record holder, if for all $n<x$, the $3n+1$ sequence produces less Chen-primes than $3x+1$ before reaching $1$.

Which is the sequence of record holders?

Answer 1

A similar sequence but for primes is A078373.

You can take the MATHEMATICA code from there and replace the prime test PrimeQ with a Chen prime test ($p$ prime and $p+2$ prime or semiprime) ChenPrimeQ, to get your sequence. For example:

ChenPrimeQ[n_] := PrimeQ[n] && (Total[FactorInteger[n + 2, 3]][[2]] <= 2);

Then I get for your sequence:

$$ 2, 3, 7, 19, 27, 109, 171, 231, 703, 937, 1263, 2919, 6919, 18451, 21867, 136359, 146599, 293199, \dots $$