I checked (b1)*b^n+1 for 2<=b<=500, (b+1)*b^n1 for 2<=b<=300 and (b+1)*b^n+1 for 2<=b<=200.
For the primes of the form (b+1)*b^n+1 with integer b>=2 and integer n>=1: For (b1)*b^n1, it is already searched in [URL]http://harvey563.tripod.com/wills.txt[/URL] for b<=2049, but one prime is missing in this website: (911)*91^5191, and the exponent of b=38 is wrong, it should be (381)*38^1362111, not (381)*38^1362211. Besides, (1281)*128^n1 has been reserved by Cruelty. The known primes with b<=500 and n>1000 are (381)*38^1362111, (831)*83^214951, (981)*98^49831, (1131)*113^2866431, (1251)*125^87391, (1881)*188^135071, (2281)*228^36951, (3471)*347^44611, (3571)*357^13191, (4011)*401^1036691, (4171)*417^210021, (4431)*443^16911, (4581)*458^468991, (4941)*494^215791. The bases b<=500 without known prime are 128 (n>1700000), 233, 268, 293, 383, 478, 488, all are checked to at least n=200000. For (b1)*b^n+1, the known primes with b<=500 and n>500 are (531)*53^960+1, (651)*65^946+1, (771)*77^828+1, (881)*88^3022+1, (1221)*122^6216+1, (1581)*158^1620+1, (1801)*180^2484+1, (1971)*197^520+1, (2481)*248^604+1, (2491)*249^1851+1, (2571)*257^1344+1, (2691)*269^1436+1, (2751)*275^980+1, (3191)*319^564+1, (3561)*356^528+1, (4341)*434^882+1. The bases b<=500 without known prime are 123 (n>100000), 202 (reserving, n>1024), 251 (n>73000), 272 (reserving, n>1024), 297 (CRUS prime), 298, 326, 328, 342 (n>100000), 347, 362, 363, 419, 422, 438 (n>100000), 452, 455, 479, 487 (n>100000), 497, 498 (CRUS prime), all are checked to at least n=1024. For (b+1)*b^n1, the known primes with b<=300 and n>500 are (63+1)*63^14831, (88+1)*88^17041, (143+1)*143^9211. The bases b<=300 without known primes are 208, 232, 282, 292, all are checked to at least n=1024. (except the case b=208, all of them are CRUS primes) For (b+1)*b^n+1, in this case this b should not = 1 (mod 3), or all numbers of the form (b+1)*b^n+1 are divisible by 3, the known primes with b<=200 (b != 1 mod 3) and n>500 are (171+1)*171^1851+1, there is no such prime with b=201 and n<=1024. 
4*5^2829891 is prime! (197802 decimal digits), [url='https://oeis.org/A046865']A046865[/url] changed, continuing.

[QUOTE=kar_bon;501959]4*5^2829891 is prime! (197802 decimal digits), [url='https://oeis.org/A046865']A046865[/url] changed, continuing.[/QUOTE]
Where is the wiki... I want the data for all Williams (1st, 2nd, 3rd and 4th) primes (and their dual) for bases b <= 32 ... (and my additional search for b = 38, 83, 88, 93 and 113 ...) 
[QUOTE=kar_bon;501959][url='https://oeis.org/A046865']A046865[/url] changed, continuing.[/QUOTE]
I don't see this change in A046865. Maybe it is still stuck in editing? Is it confirmed that there are no intervening terms between 15393 and 282989? If so, 282989 can be added, but if not, then you should add a comment instead: "282989 is also a member of this sequence". 
Edits confirmed and I've checked the whole range again, continuing.

Always press 'these edits are ready for review'. It's a gotcha in the OEIS Wiki.
(It was not always on the Wiki engine, actually.) 
[QUOTE=Batalov;502892]Always press 'these edits are ready for review'. It's a gotcha in the OEIS Wiki.[/QUOTE]
Yes, overviewd this button on the bottom but got a mail after 7 days without doing so. 
Found:
4*5^4984831 is prime! (348426 decimal digits) 4*5^5042211 is prime! (352436 decimal digits) Currently at n=695k, OEIS updated. 
[url='https://primes.utm.edu/primes/page.php?id=125947']4*5^7546111[/url] is prime: 527452 decimal digits.

[url='https://primes.utm.edu/primes/page.php?id=126245']4*5^8647511[/url] is prime: 604436 decimal digits.
I've not updated the OEIS seq with the last two primes, will do after reaching n=1M. 
4*5^n1 completed to n=1M and releasing
 no more primes found  OEIS / Wiki updated  S.Harvey informed 
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