primes of ther form 19*3*2^k+ or -1
 

I found only three primes of this form up to k=10.000. (k=2,8,18)


Are they finite?

There should be infinitely many. You can look at the residue classes and do an infinite product to guess how sparse they will be.

Looks like the OP missed quite a few. If the intent was to find exponents on [i]both[/i] lists, the exponent 10 is missing and the exponent 18 is wrong.

From the [url=http://www.prothsearch.com/riesel1.html]List of primes k*2[sup]n[/sup] + 1 for k < 300[/url] we have for k = 19*3 = 57,
[quote]57 2, 3, 7, 8, 10, 16, 18, 19, 40, 48, 55, 90, 96, 98, 190, 398, 456, 502, 719, 1312, 1399, 1828, 6723, 6816, 10680, 12592, 20742, [b]25010[/b], 26838, 29623, 45435, 52783, 70950, 89691, 111691, 114400, 136152, 145183, [b]146223[/b], 177459, 212908, 300910, 342151, 360447, 382156, 411635, 442948, 519862, 519892, 975036, 1158942, 1181438, 1756702, 2033643 [2100000] 2492031 L1230, [b]2747499[/b] L3514, 2765963 L3262[/quote] The exponents in bold give factors of Fermat numbers.

From the [url=http://www.prothsearch.com/riesel2.html]List of primes k*2[sup]n[/sup] - 1 for k < 300[/url], again for k = 19*3 = 57,
[quote]57 1, 2, 4, 5, 8, 10, 20, 22, 25, 26, 32, 44, 62, 77, 158, 317, 500, 713, 1657, 1790, 2761, 2794, 3704, 4174, 6772, 14348, 16132, 16160, 16766, 21097, 29125, 40094, 44824, 49006, 67585, 74650, 94798, 162538, 173585, 239854 [400000] 450430 L171, 565994 L261, 707245 L384, 839446 L80, 1098272 L260, 1110980 L121, 1486214 L1828, 2103370 L2055, 2639528 L2484, 3339932 L3519[/quote]
(The L's denote "proof codes.")