A108974 as an irregular array. Row k gives odd primes for which the multiplicative order of two is exactly k.
See A086251 for row lengths.

Produced with PARI/GP:
  lastrow=300;for(k=2,lastrow,print(k,": ",select(p -> p!=2 && znorder(Mod(2,p),k)==k, factor(2^k-1)[,1]~)))
Code not optimized (for example, when factoring 2^202-1, we do not utilize that we already did 2^101-1 earlier).


2: [3]
3: [7]
4: [5]
5: [31]
6: []
7: [127]
8: [17]
9: [73]
10: [11]
11: [23, 89]
12: [13]
13: [8191]
14: [43]
15: [151]
16: [257]
17: [131071]
18: [19]
19: [524287]
20: [41]
21: [337]
22: [683]
23: [47, 178481]
24: [241]
25: [601, 1801]
26: [2731]
27: [262657]
28: [29, 113]
29: [233, 1103, 2089]
30: [331]
31: [2147483647]
32: [65537]
33: [599479]
34: [43691]
35: [71, 122921]
36: [37, 109]
37: [223, 616318177]
38: [174763]
39: [79, 121369]
40: [61681]
41: [13367, 164511353]
42: [5419]
43: [431, 9719, 2099863]
44: [397, 2113]
45: [631, 23311]
46: [2796203]
47: [2351, 4513, 13264529]
48: [97, 673]
49: [4432676798593]
50: [251, 4051]
51: [103, 2143, 11119]
52: [53, 157, 1613]
53: [6361, 69431, 20394401]
54: [87211]
55: [881, 3191, 201961]
56: [15790321]
57: [32377, 1212847]
58: [59, 3033169]
59: [179951, 3203431780337]
60: [61, 1321]
61: [2305843009213693951]
62: [715827883]
63: [92737, 649657]
64: [641, 6700417]
65: [145295143558111]
66: [67, 20857]
67: [193707721, 761838257287]
68: [137, 953, 26317]
69: [10052678938039]
70: [281, 86171]
71: [228479, 48544121, 212885833]
72: [433, 38737]
73: [439, 2298041, 9361973132609]
74: [1777, 25781083]
75: [100801, 10567201]
76: [229, 457, 525313]
77: [581283643249112959]
78: [22366891]
79: [2687, 202029703, 1113491139767]
80: [4278255361]
81: [2593, 71119, 97685839]
82: [83, 8831418697]
83: [167, 57912614113275649087721]
84: [1429, 14449]
85: [9520972806333758431]
86: [2932031007403]
87: [4177, 9857737155463]
88: [353, 2931542417]
89: [618970019642690137449562111]
90: [18837001]
91: [911, 112901153, 23140471537]
92: [277, 1013, 1657, 30269]
93: [658812288653553079]
94: [283, 165768537521]
95: [191, 420778751, 30327152671]
96: [193, 22253377]
97: [11447, 13842607235828485645766393]
98: [4363953127297]
99: [199, 153649, 33057806959]
100: [101, 8101, 268501]
101: [7432339208719, 341117531003194129]
102: [307, 2857, 6529]
103: [2550183799, 3976656429941438590393]
104: [858001, 308761441]
105: [29191, 106681, 152041]
106: [107, 28059810762433]
107: [162259276829213363391578010288127]
108: [246241, 279073]
109: [745988807, 870035986098720987332873]
110: [2971, 48912491]
111: [321679, 26295457, 319020217]
112: [5153, 54410972897]
113: [3391, 23279, 65993, 1868569, 1066818132868207]
114: [571, 160465489]
115: [14951, 4036961, 2646507710984041]
116: [107367629, 536903681]
117: [937, 6553, 86113, 7830118297]
118: [2833, 37171, 1824726041]
119: [239, 20231, 62983048367, 131105292137]
120: [4562284561]
121: [727, 1786393878363164227858270210279]
122: [768614336404564651]
123: [3887047, 177722253954175633]
124: [5581, 8681, 49477, 384773]
125: [269089806001, 4710883168879506001]
126: [77158673929]
127: [170141183460469231731687303715884105727]
128: [274177, 67280421310721]
129: [11053036065049294753459639]
130: [131, 409891, 7623851]
131: [263, 10350794431055162386718619237468234569]
132: [312709, 4327489]
133: [163537220852725398851434325720959]
134: [7327657, 6713103182899]
135: [271, 348031, 49971617830801]
136: [354689, 2879347902817]
137: [32032215596496435569, 5439042183600204290159]
138: [139, 168749965921]
139: [5625767248687, 123876132205208335762278423601]
140: [7416361, 47392381]
141: [4375578271, 646675035253258729]
142: [56409643, 13952598148481]
143: [724153, 158822951431, 5782172113400990737]
144: [577, 487824887233]
145: [2679895157783862814690027494144991]
146: [1753, 1795918038741070627]
147: [2741672362528725535068727]
148: [149, 593, 184481113, 231769777]
149: [86656268566282183151, 8235109336690846723986161]
150: [1133836730401]
151: [18121, 55871, 165799, 2332951, 7289088383388253664437433]
152: [1217, 148961, 24517014940753]
153: [919, 75582488424179347083438319]
154: [617, 78233, 35532364099]
155: [311, 11471, 73471, 4649919401, 18158209813151]
156: [313, 1249, 3121, 21841]
157: [852133201, 60726444167, 1654058017289, 2134387368610417]
158: [201487636602438195784363]
159: [6679, 13960201, 540701761, 229890275929]
160: [414721, 44479210368001]
161: [1289, 3188767, 45076044553, 14808607715315782481]
162: [163, 135433, 272010961]
163: [150287, 704161, 110211473, 27669118297, 36230454570129675721]
164: [10169, 181549, 12112549, 43249589]
165: [2048568835297380486760231]
166: [499, 1163, 2657, 155377, 13455809771]
167: [2349023, 79638304766856507377778616296087448490695649]
168: [3361, 88959882481]
169: [4057, 6740339310641, 3340762283952395329506327023033]
170: [26831423036065352611]
171: [93507247, 3042645634792541312037847]
172: [173, 101653, 500177, 1759217765581]
173: [730753, 1505447, 70084436712553223, 155285743288572277679887]
174: [96076791871613611]
175: [39551, 60816001, 535347624791488552837151]
176: [229153, 119782433, 43872038849]
177: [184081, 27989941729, 9213624084535989031]
178: [179, 62020897, 18584774046020617]
179: [359, 1433, 1489459109360039866456940197095433721664951999121]
180: [181, 54001, 29247661]
181: [43441, 1164193, 7648337, 7923871097285295625344647665764672671]
182: [224771, 1210483, 25829691707]
183: [367, 55633, 37201708625305146303973352041]
184: [291280009243618888211558641]
185: [1587855697992791, 7248808599285760001152755641]
186: [529510939, 2903110321]
187: [707983, 1032670816743843860998850056278950666491537]
188: [3761, 7484047069, 140737471578113]
189: [1560007, 207617485544258392970753527]
190: [2281, 3011347479614249131]
191: [383, 7068569257, 39940132241, 332584516519201, 87274497124602996457]
192: [18446744069414584321]
193: [13821503, 61654440233248340616559, 14732265321145317331353282383]
194: [971, 1553, 31817, 1100876018364883721]
195: [134304196845099262572814573351]
196: [197, 19707683773, 4981857697937]
197: [7487, 26828803997912886929710867041891989490486893845712448833]
198: [5347, 242099935645987]
199: [164504919713, 4884164093883941177660049098586324302977543600799]
200: [401, 340801, 2787601, 3173389601]
201: [1609, 22111, 87449423397425857942678833145441]
202: [845100400152152934331135470251]
203: [136417, 121793911, 11348055580883272011090856053175361113]
204: [409, 3061, 13669, 1326700741]
205: [2940521, 70171342151, 3655725065508797181674078959681]
206: [415141630193, 8142767081771726171]
207: [79903, 634569679, 2232578641663, 42166482463639]
208: [78919881726271091143763623681]
209: [94803416684681, 1512348937147247, 5346950541323960232319657]
210: [211, 664441, 1564921]
211: [15193, 60272956433838849161, 3593875704495823757388199894268773153439]
212: [15358129, 586477649, 1801439824104653]
213: [66457, 2849881972114740679, 4205268574191396793]
214: [643, 84115747449047881488635567801]
215: [1721, 731516431, 514851898711, 297927289744047764444862191]
216: [33975937, 138991501037953]
217: [5209, 62497, 6268703933840364033151, 378428804431424484082633]
218: [104124649, 2077756847362348863128179]
219: [3943, 671165898617413417, 4815314615204347717321]
220: [415878438361, 3630105520141]
221: [1327, 2365454398418399772605086209214363458552839866247069233]
222: [3331, 17539, 107775231312019]
223: [18287, 196687, 1466449, 2916841, 1469495262398780123809, 596242599987116128415063]
224: [449, 2689, 183076097, 358429848460993]
225: [115201, 617401, 1348206751, 13861369826299351]
226: [227, 48817, 636190001, 491003369344660409]
227: [26986333437777017, 7992177738205979626491506950867720953545660121688631]
228: [131101, 160969, 275415303169]
229: [1504073, 20492753, 59833457464970183, 467795120187583723534280000348743236593]
230: [691, 1884103651, 345767385170491]
231: [463, 4982397651178256151338302204762057]
232: [59393, 82280195167144119832390568177]
233: [1399, 135607, 622577, 116868129879077600270344856324766260085066532853492178431]
234: [5302306226370307681801]
235: [2391314881, 72296287361, 73202300395158005845473537146974751]
236: [1181, 3541, 157649, 174877, 5521693, 104399276341]
237: [1423, 49297, 23728823512345609279, 31357373417090093431]
238: [823679683, 143162553165560959297]
239: [479, 1913, 5737, 176383, 134000609, 7110008717824458123105014279253754096863768062879]
240: [394783681, 46908728641]
241: [22000409, 160619474372352289412737508720216839225805656328990879953332340439]
242: [117371, 11054184582797800455736061107]
243: [487, 16753783618801, 192971705688577, 3712990163251158343]
244: [733, 1709, 3456749, 368140581013, 667055378149]
245: [1471, 252359902034571016856214298851708529738525821631]
246: [739, 165313, 13194317913029593]
247: [15809, 6459570124697, 402004106269663, 1282816117617265060453496956212169]
248: [290657, 3770202641, 1141629180401976895873]
249: [1621324657, 8241594690167137359552274418432855740327]
250: [229668251, 5519485418336288303251]
251: [503, 54217, 178230287214063289511, 61676882198695257501367, 12070396178249893039969681]
252: [40388473189, 118750098349]
253: [4103188409, 199957736328435366769577, 44667711762797798403039426178361]
254: [56713727820156410577229101238628035243]
255: [106591, 949111, 5702451577639775545838643151]
256: [59649589127497217, 5704689200685129054721]
257: [535006138814359, 1155685395246619182673033, 374550598501810936581776630096313181393]
258: [1033, 1591582393, 15686603697451]
259: [2499285769, 21234370960880098806027750185552713706866970578963970119]
260: [521, 51481, 34110701, 108140989558681]
261: [328017025014102923449988663752960080886511412965881]
262: [1049, 4744297, 182331128681207781784391813611]
263: [23671, 13572264529177, 120226360536848498024035943, 383725126655170964501315730676446647]
264: [7393, 1761345169, 98618273953]
265: [29324808311, 197748738449921, 36614110124735294634435619027766763481]
266: [4523, 106788290443848295284382097033]
267: [78903841, 28753302853087, 24124332437713924084267316537353]
268: [269, 15152453, 42875177, 2559066073, 9739278030221]
269: [13822297, 68625988504811774259364670661552948915363901845035416371912463477873783063]
270: [811, 15121, 385838642647891]
271: [15242475217, 248927757868131890277330541567820045256364273970773286542188386932989391]
272: [383521, 2368179743873, 373200722470799764577]
273: [108749551, 4093204977277417, 86977595801949844993]
274: [1097, 15619, 32127963626435681, 105498212027592977]
275: [382027665134363932751, 4074891477354886815033308087379995347151]
276: [5415624023749, 70334392823809]
277: [1121297, 31133636305610209482201109050392404721, 6955979459776540052280934851589652278783]
278: [4506937, 51542639524661795300074174250365699]
279: [16183, 34039, 1437967, 833732508401263, 2034439836951867299888617]
280: [84179842077657862011867889681]
281: [80929, 48009215293052652841860443273079338843737271906291675944391068955229998769420319]
282: [1681003, 35273039401, 111349165273]
283: [9623, 68492481833, 23579543011798993222850893929565870383844167873851502677311057483194673]
284: [569, 148587949, 4999465853, 5585522857, 472287102421]
285: [1491477035689218775711, 25349242986637720573561]
286: [2003, 6156182033, 10425285443, 15500487753323]
287: [17137716527, 51954390877748655744256192963206220919272895548843817842228913]
288: [1153, 6337, 38941695937, 278452876033]
289: [12761663, 179058312604392742511009, 3320934994356628805321733520790947608989420068445023]
290: [7553921, 999802854724715300883845411]
291: [272959, 2065304407, 5434876633, 1170711644777651877659556633665719]
292: [293, 9929, 649301712182209, 9444732965601851473921]
293: [40122362455616221971122353, 396645227028138890415611220710757921643910743103031701971222447]
294: [748819, 26032885845392093851]
295: [4721, 132751, 5794391, 128818831, 3812358161, 452824604065751, 4410975230650827973711]
296: [20988936657440586486151264256610222593863921]
297: [8950393, 170886618823141738081830950807292771648313599433]
298: [1193, 650833, 38369587, 7984559573504259856359124657]
299: [599, 9341359, 14718679249, 13444476836590589479, 51441563151591093599, 260242449712509916159]
300: [1201, 63901, 13334701, 1182468601]

/JeppeSN