What's the deal with k=444,637? Why is it being tested for such large N values?

That number is still unproven. The full list taken from another post on this page is :--

2293, 9221, 23669, 31859, 38473, 40597, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 146561, 161669, 192971, 206039, 206231, 215443, 226153, 234343, 245561, 250027, 252191, 273809, 304207, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 398023, 402539, 409753, 444637, 470173, 474491, 477583, 485557, 494743, 502573

So there are 6 even higher still to be tested.
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Member team AUSTRALIA
My lucky number is 9291*2^1085585+1

I get those kind of WU too.

However that's quite strange.

For k=444637 and k=342847 we are checking very large value of n, and probably not sequentially. Why this? It is better to check lower values, which are more probable to be prime and also are less cpu-consuming. Furthermore if we found a prime for larger n, we still have to check smaller n to find the smallest n for that k.

This is a non-sense.
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Furthermore if we found a prime for larger n, we still have to check smaller n to find the smallest n for that k

and probably not sequentially

As discussed in the doublecheck topic there are some advantages on knowing what is the smallest n for that k (for statistical analysis about primes), even if this is irrelevant for the Riesel conjecture.

And, of course, WU are not completed sequentially (it depends on each user machine), but they *should* be started sequentially. I *doubt* that PrimeGrid has sent all WU up to n=9M for that two values of k.
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Sorry. I guess I wasn't clear enough. I was getting WUs that had n=9M+. Some of the WUs were only 6M or 7M, but still, that is much larger than the typical leading edge of any other k value; ~5.4M

I was just wondering why k=444,637 was special. In fact, I'm still wondering why I got such lengthy work units. We weren't given enough time to finish the 9M on time.