850612425 filled gap
 

filled gap upto 11k:
6, 7, 8, 10, 11, 17, 20, 25, 35, 44, 57, 66, 71, 80, 84, 104, 109, 119, 120, 144, 164, 171, 176, 188, 218, 224, 225, 250, 257, 260, 269, 273, 324, 325, 360, 400, 409, 414, 453, 468, 500, 551, 619, 627, 717, 778, 786, 1071, 1084, 1271, 1287, 1296, 1336, 1349, 1450, 1599, 1687, 1765, 1898, 2355, 2369, 2920, 3209, 4189, 4598, 4634, 4931, 5237, 5813, 5851, 6409, 6475, 6608, 6806, 8793, 9814, 10649.

i count only 99 primes upto 275965 (not 100!).
6 Sophie Germain and one twin at n=84

k=1311455145 resolved "(92 primes 0-50k)"
 

prime n: 2, 3, 5, 10, 13, 14, 19, 20, 27, 28, 35, 38, 40, 49, 56, 65, 81, 91, 102, 103, 105, 119, 127, 161, 209, 210, 231, 233, 270, 318, 399, 424, 427, 487, 503, 524, 525, 548, 572, 589, 593, 657, 664, 781, 842, 853, 1105, 1238, 1239, 1268, 1398, 1520, 1596, 1733, 2247, 2261, 2375, 2454, 3128, 3717, 3721, 4433, 4510, 5471, 5554, 5700, 6633, 6804, 6974, 7467, 7796, 7979, 8794, 9019, 12772, 12776, 14363, 15191, 19394, 20967, 23294, 24322, 25958, 28091, 29747, 29811, 34527, 35991, 40807, 42511, 42987, 46456.

92 primes upto 50k.
8 Sophie Germain and 3 twins at n=14, 19 and 525

k=1346178405 (98 primes 0-50k)
 

n prime upto n=50k:
1, 3, 5, 10, 12, 17, 18, 21, 22, 27, 34, 36, 38, 44, 46, 48, 54, 56, 67, 76, 78, 83, 84, 88, 90, 98, 99, 103, 105, 106, 111, 146, 156, 168, 170, 212, 249, 256, 260, 277, 295, 300, 320, 323, 385, 436, 502, 539, 575, 602, 603, 1025, 1054, 1178, 1185, 1339, 1340, 1594, 1604, 1870, 1966, 2585, 2732, 2808, 2855, 2872, 3090, 4134, 4785, 4874, 5947, 6069, 6717, 7174, 7476, 7664, 8436, 9074, 9566, 10840, 11766, 15394, 15498, 16674, 18344, 19272, 20723, 24395, 24747, 24929, 25118, 29696, 32257, 36915, 37492, 38179, 41470, 43483.

7 Sophie Germain and one twin at n=27.

so 99 primes at all with a gap from n=50k to 200k.

k=1875 and k=1995
 

Primes for k = 1875, 10k to 100k
10089, 16291, 18757, 20244, 20331, 25005, 25281, 25943, 27252, 32943, 35199, 44218, 64672, 64907, 80627, 85745, 86184, 88381, 89250, 99362

Primes for k = 1995, 10k to 100k
10566, 13882, 15482, 15585, 18022, 19409, 20654, 23375, 24412, 26842, 34369, 36104, 36594, 39039, 42840, 43564, 56962, 74223, 81227, 81349, 86341

'Reserving' these two k please.

k=20152491645
 

I have filled in the gap from n=100-300k. This makes 137 primes to n=500k.

[CODE]112108, 113125, 113562, 114440, 120649, 122777, 147961, 180723,
184473, 197163, 208554, 280462, 288739[/CODE]

k=52654604145
 

I have filled the gap between n=100-300k. This makes 137 primes to n=500k.

[CODE]103402, 114420, 137008, 139685, 149831, 152588, 156808, 173837,
174967, 175138, 201931, 225859, 227757, 285607, 288055, 295269[/CODE]

milestone report for k=151515
 

k=151515 tested till n=1.1M, switching to sieve for a moment :cool:

more small primes and twins
 

don't want to overload the thread so only stats without explicit n's.
(data's are: k: range, # primes, # Sophie Germain, twins)

small primes and twins
1015239225: 0-50k, 66 primes, 6 SG(=CC7), 4 twins: 4,5,142,235
1386560175: 0-50k, 92 primes, 5 SG, 6 twins: 1,3,5,15,61,263
1501946985: 0-50k, 100 primes, 6 SG, 6 twins: 10,16,21,48,80,154
1533677145: 0-50k, 91 primes, 4 SG, 3 twins: 4,27,33
1748348745: 0-50k, 98 primes, 4 SG, 2 twins: 36,489

only twins: (checked listed prime n's only)
20152491645: none
(a k with nash=7452 and 137 primes upto 500k without any twin. wow!!! quite remarkable)
20887630335: n=988
28509385905: n=4,15,78,330
31194316725: n=7,19,30,97
34805003805: n=15,21,24,42,69,78,414
37850187375: n=5,13,20,64,100,130
38101088025: none
38805158535: n=5,77
52654604145: n=201

all data are available with next update. more to come.

Status from Templus on k=25935
 

Below is a status that I got from Templus on k=25935. I'll inform him to post them here in the future.

[quote=Templus]

I have doublechecked the range by Fatphil until n=220k for k=25935*2^n-1, I am now able to test it further.

Kind Regards,

Templus (Ingmar)
[/quote]

k=1801 tested to 100k:
10371
20445
38443

more small primes and twins
 

1041415155: 0-300k, 28 primes, no SG, no twin
2028893625: 0-50k, 98 primes, 5 SG, twins n=2,11
2037910875: 0-50k, 95 primes, 9 SG, twin n=29
2241069285: 0-50k, 94 primes, 4 SG, twins n=15,54,72
2481187995: 0-50k, 91 (not 92) primes, 5 SG, twins n=22,52,97
2641908225: 0-50k, 92 primes, 6 SG, twin n=37,1051
2904473715: 0-50k, 93 primes, 6 SG, no twin
3289888875: 0-50k, 95 primes, 5 SG, twin n=3,7,10,25
3870706125: 0-50k, 92 primes, 5 SG, twin n=119,1633

filling gaps:
2317: 10k-200k, 2 primes n=14657, 17017