100 primes
For the groups interest, the best +1 number to get to a hundred primes got there with n=3258 with k = 1108828374241*M(59), where M(59) is a certain multiple of small primes p up to 59, excluding those with order base 2 of less than p1. This is a Payam number, par excellence.
It may well be that the  series produced better, but I don't know for sure. In any case there are better numbers out there, although Phil Carmody thought that this was close to the ideal number at to that (i.e. 100 primes) range.
For your interest, this number was checked up to n=100000 with "only" 140 primes, so we stopped checking it in earnest  ther are plenty with greater than that at that level, but software does not allow for fast checking of theses k, because k is rather large.
Regards
Robert Smith
Ps this series is prime at n= (note the huge gap at 44957544)
3
11
13
14
18
19
23
25
26
29
31
42
45
50
64
65
71
86
98
101
116
119
134
191
194
195
212
257
259
269
276
284
285
294
307
324
396
403
406
420
447
449
459
480
486
540
545
602
607
625
654
659
703
742
780
827
828
838
848
867
923
960
976
982
1005
1011
1021
1037
1049
1081
1100
1130
1145
1192
1254
1327
1427
1488
1548
1599
1647
1663
1684
1818
1844
1861
1880
1892
1946
1951
1971
2044
2130
2150
2227
2328
2393
3215
3224
3258 no 100
3289
3405
3436
3450
3722
3833
4172
4227
4337
4495
7544
8666
9037
9263
9758
10493
10628
12817
12959
15351
15762
16529
18995
19249
21117
22601
25769
27486
32648
33107
38099
40751
52586
53046
53060
53105
54241
60535
75013
85111
102912
126708
