An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

I have found a significant number of polynomials of this type with $T\ge 24$ but nothing with $T>40$.

The prime $k$-tuples conjecture suggests that there should be examples with $T$ arbitrarily large since $2n^2$ and $n^2+n$ omit some residue classes for every prime. For example, for $n=0,1,\ldots,9$ the differences $n^2+n-(0^2+0)$ are $(0,2,6,12,20,30,42,56,72,90)$ which forms an admissible 10-tuple of differences for primes. So we may expect there are infinitely many sets of primes with these differences, for example there are starting at,

$$11,17,41,844427,51448361,86966771,122983031,180078317$$

(Edit: See A191456.) So to generate these polynomials with $T\ge 10$ we can just look for these, e.g. $n^2+n+41$, $n^2+n+51448361$ and $n^2+n+180078317$ all work, the first and last have $T>10$. Similarly the differences for $n=0,1,\ldots,39$ form an admissible 40-tuple so there should be larger primes $q$ such that $n^2+n+q$ has $T\ge 40$, but they will be tough to find with brute force.

Here are some with $T\ge 27$. Here $\operatorname{sqfr}(d)$ is the square-free part of the discriminant. $$ \begin{array}{|cccccc|} \hline Type & P(n) & T & \operatorname{sqfr}(d) & h(d) \\ \hline I & n^2+n-1354363 & 29 & \color{blue}{5417453} & 4 \\ I & 2(n^2+n)-177953 & 27 & 355907 & 2 \\ I & 3(n^2+n)-675299 & 34 & 8103597 & 6 \\ I & 3(n^2+n)-122957 & 30 & 1475493 & 2 \\ I & 5(n^2+n)-65063 & 27 & 1301285 & 4 \\ I & 5(n^2+n)-611903 & 27 & 12238085 & 4 \\ I & 5(n^2+n-6)\color{green}{-281837} & 27 & \color{green}{5637365} & 2 \\ I & 9(n^2+n)-90071 & 27 & 360293 & 1 \\ I & 9(n^2+n)-867551 & 27 & 3470213 & 3 \\ I & 11(n^2+n)-258113 & 27 & 11357093 & 1 \\ I & 12(n^2+n)-236111 & 27 & 708342 & 4 \\ I & 15(n^2+n)-157147 & 27 & 9429045 & 8 \\ I & 22(n^2+n)-330271 & 28 & 7266083 & 8 \\ I & 22(n^2+n)-10273 & 28 & 226127 & 4 \\ I & 35(n^2+n)+6283 & 24 & -878395 & 92 \\ I & 38(n^2+n)-9287 & 34 & 353267 & 4 \\ I & 41(n^2+n)-33023 & 29 & \color{blue}{5417453} & 4 \\ I & 45(n^2+n)-1322611 & 29 & 26452445 & 2 \\ I & 125(n^2+n)\color{green}{-281837} & 27 & \color{green}{5637365} & 2 \\ I & 175(n^2+n)-333103 & 28 & 9328109 & 1 \\ I & 210(n^2+n) - 71899 & 29 & 15109815 & 32 \\ \hline II & 2n^2-181 & 28 & 362 & 2 \\ II & 6n^2-140897 & 33 & 845382 & 6 \\ II & 14n^2-85093 & 28 & 1191302 & 2 \\ II & 22n^2-20051 & 27 & 441122 & 2 \\ II & 30n^2-176399 & 27 & 5291970 & 8 \\ II & 38n^2-856759 & 28 & 32556842 & 2 \\ II & 42n^2-153779 & 28 & 6458718 & 8 \\ II & 258n^2+3331 & 27 & -859398 & 240 \\ \hline \end{array} $$

$n^2+n-1354363$ also has another run of 18 primes and $14n^2-85093$ another run of 17.

As per the request in the comments, here are a few more pairs of polynomials with matching $\operatorname{sqfr}(d)$ (one is repeated from above). $$ \begin{array}{|cccc|} \hline P(n) & T & \operatorname{sqfr}(d) & h(d) \\ \hline 3(n^2+n-2)-58111 & 20 & 697413 & 4 \\ 3^3(n^2+n)-58111 & 20 & 697413 & 4 \\ \hline 3(n^2+n-2)-92893 & 20 & 1114797 & 2 \\ 3^3(n^2+n)-92893 & 18 & 1114797 & 2 \\ \hline 3(n^2+n-2)-1070633 & 16 & 12847677 & 2 \\ 3^3(n^2+n)-1070633 & 18 & 12847677 & 2 \\ \hline 5(n^2+n-6)-281837 & 27 & 5637365 & 2 \\ 5^3(n^2+n)-281837 & 27 & 5637365 & 2 \\ \hline 5(n^2+n-6)-1076687 & 13 & 21534365 & 12 \\ 5^3(n^2+n)-1076687 & 19 & 21534365 & 12 \\ \hline 7(n^2+n-12)-112417 & 18 & 3150077 & 1 \\ 7^3(n^2+n)-112417 & 18 & 3150077 & 1 \\ \hline 7(n^2+n-12)-214519 & 18 & 6008933 & 14 \\ 7^3(n^2+n)-214519 & 16 & 6008933 & 14\\ \hline \end{array} $$