Special n

[kar_bon]

Here is the place to post your search-results for special exponents.
It's like a Constant n Search but i think it's worth an own thread.

For example:
L.Zhou searches for General Mersenne Primes, a prime with an exponent like k=1.
So 755061*2^756839-1 is a General Mersenne Prime for M32 (= 2^756839-1).

Next:
He also searches for n=2^2^19 = 524288 so 428079*2^524288-1 is the first prime.

Another:
The next exponent is the 29th Fibonacci-Number (f(n)=f(n-1)+f(n-2) with f(0)=0, f(1)=1) and he found 3681*2^514229-1 as first prime!

Only some examples so you can imagine when posting here.

I'll expand the examples with some results when ready.

[Harvey563]

Here are some old searches (2002) where n is one less than a power of 2

k*2^(2^15-1)-1 to 60k [56083]

k*2^(2^16-1)-1 to 300k [130141, 133471, 202881,
205851, 247765, 280731, 281383, 287769, 291771]

k*2^(2^17-1)-1 to 10000000
[49905, 105915, 108141, 162589, 290173,
323721, 335323, 336289, 402033, 402933, 406875,
449259, 756901, 873495, 939603, 1042009,
1164781, 1167529, 1426411, 1444729, 1519773,
1541373, 1550395, 1747023, 1829443, 1881643,
1905385, 2086119, 2114689, 2153089, 2185195,
2204653, 2317309, 2408185, 2413405, 2433045,
2462175, 2552121, 2558013, 2588301, 2637565,
2646601, 2708103, 2744811, 2825209, 2893609,
2974281, 3004861, 3028293, 3081639, 3104061,
3159709, 3347109, 3390919, 3493303, 3609655,
4063821, 4077655, 4164993, 4183761, 4193803,
4385313, 4541979, 4745925, 4845303, 5025223, 5042331,
5049103, 5086663, 5162859, 5475373, 5510941,
5517849, 5646285, 5750739, 6033223, 6308343,
6358651, 6396291, 6431719, 6854983, 6870435,
6880111, 7041697, 7044289, 7097169, 7102173,
7331703, 7356265, 7430145, 7434021, 7496643,
7602349, 7712515, 7984863, 8036533, 8078925, 8150985,
8189923, 8206323, 8314165, 8350119, 8460349,
8477779, 8513935, 8564655, 8739085, 8785269,
9008209, 9009819, 9139815, 9236929, 9361651,
9435853, 9534301, 9569169, 9661771, 9930265]

Harvey563