Because I was interested how sieving worked and also wanted to test Proth20 (proth testing for GPU), I started looking at 1281979 * 2^n +1.

That k is large enough not to get into PG's various Proth subprojects. Also it's prime, which I find interesting since it means there can be a lot of small factors in 1281979 * 2^n + 1 so sieving will be quite efficient. Of course it also means the probability for smaller primes is low, so it evens out.

And it's my birthday. ;)
**Sieving**
Range: 100 000 < n < 4 100 000

Factors: p < 290e12

I sieved with sr1sieve on a Pentium G870 which took about 6 months. I stopped when finding a factor took about 2 hours. 66272 candidates remained.

I know now I should have sieved to much higher n, but I didn't know that when I began.

**Primality testing**
Software is LLR2, n < 1 200 000 was tested on the Pentium G870. Now I switched to a Ryzen 9 3900X. For n = 1 200 000 a test takes 460 s, for n = 2 300 000 it takes 1710 s, showing nicely how well the approximation "double the digits, quadruple the time" can be applied. At least as long as FFT and L3 sizes match.

**Results**
Code:

For n < 1 000 000
k = digits
2 7 (not a Proth)
142 49
202 67
242 79
370 118
578 180
614 191
754 233
6430 1942
7438 2246
10474 3159
11542 3481
45022 13559
46802 14095
70382 21194
74938 22565
367310 110578
485014 146010

Next milestone is n = 1 560 000 at which point any prime would be large enough to enter Caldwell's Top 5000.