Algebraic factors:
Cunningham numbers: (Phi is the cyclotomic polynomial)
2^r-1 = prod_{d dividing r}(Phi_d(2))
2^r+1 = prod_{d dividing 2*r, d not dividing r}(Phi_d(2))
Period 2:
9*2^(2*r)-1 = (3*2^r-1) * (3*2^r+1)
25*2^(2*r)-1 = (5*2^r-1) * (5*2^r+1)
49*2^(2*r)-1 = (7*2^r-1) * (7*2^r+1)
81*2^(2*r)-1 = (9*2^r-1) * (9*2^r+1)
121*2^(2*r)-1 = (11*2^r-1) * (11*2^r+1)
169*2^(2*r)-1 = (13*2^r-1) * (13*2^r+1)
225*2^(2*r)-1 = (15*2^r-1) * (15*2^r+1)
289*2^(2*r)-1 = (17*2^r-1) * (17*2^r+1)
361*2^(2*r)-1 = (19*2^r-1) * (19*2^r+1)
441*2^(2*r)-1 = (21*2^r-1) * (21*2^r+1)
529*2^(2*r)-1 = (23*2^r-1) * (23*2^r+1)
625*2^(2*r)-1 = (25*2^r-1) * (25*2^r+1)
729*2^(2*r)-1 = (27*2^r-1) * (27*2^r+1)
841*2^(2*r)-1 = (29*2^r-1) * (29*2^r+1)
961*2^(2*r)-1 = (31*2^r-1) * (31*2^r+1)
1089*2^(2*r)-1 = (33*2^r-1) * (33*2^r+1)
1225*2^(2*r)-1 = (35*2^r-1) * (35*2^r+1)
1369*2^(2*r)-1 = (37*2^r-1) * (37*2^r+1)
1521*2^(2*r)-1 = (39*2^r-1) * (39*2^r+1)
1681*2^(2*r)-1 = (41*2^r-1) * (41*2^r+1)
1849*2^(2*r)-1 = (43*2^r-1) * (43*2^r+1)
2025*2^(2*r)-1 = (45*2^r-1) * (45*2^r+1)
2209*2^(2*r)-1 = (47*2^r-1) * (47*2^r+1)
2401*2^(2*r)-1 = (49*2^r-1) * (49*2^r+1)
2601*2^(2*r)-1 = (51*2^r-1) * (51*2^r+1)
2809*2^(2*r)-1 = (53*2^r-1) * (53*2^r+1)
3025*2^(2*r)-1 = (55*2^r-1) * (55*2^r+1)
3249*2^(2*r)-1 = (57*2^r-1) * (57*2^r+1)
3481*2^(2*r)-1 = (59*2^r-1) * (59*2^r+1)
3721*2^(2*r)-1 = (61*2^r-1) * (61*2^r+1)
3969*2^(2*r)-1 = (63*2^r-1) * (63*2^r+1)
4225*2^(2*r)-1 = (65*2^r-1) * (65*2^r+1)
4489*2^(2*r)-1 = (67*2^r-1) * (67*2^r+1)
4761*2^(2*r)-1 = (69*2^r-1) * (69*2^r+1)
5041*2^(2*r)-1 = (71*2^r-1) * (71*2^r+1)
5329*2^(2*r)-1 = (73*2^r-1) * (73*2^r+1)
5625*2^(2*r)-1 = (75*2^r-1) * (75*2^r+1)
5929*2^(2*r)-1 = (77*2^r-1) * (77*2^r+1)
6241*2^(2*r)-1 = (79*2^r-1) * (79*2^r+1)
6561*2^(2*r)-1 = (81*2^r-1) * (81*2^r+1)
6889*2^(2*r)-1 = (83*2^r-1) * (83*2^r+1)
7225*2^(2*r)-1 = (85*2^r-1) * (85*2^r+1)
7569*2^(2*r)-1 = (87*2^r-1) * (87*2^r+1)
7921*2^(2*r)-1 = (89*2^r-1) * (89*2^r+1)
8281*2^(2*r)-1 = (91*2^r-1) * (91*2^r+1)
8649*2^(2*r)-1 = (93*2^r-1) * (93*2^r+1)
9025*2^(2*r)-1 = (95*2^r-1) * (95*2^r+1)
9409*2^(2*r)-1 = (97*2^r-1) * (97*2^r+1)
9801*2^(2*r)-1 = (99*2^r-1) * (99*2^r+1)
10201*2^(2*r)-1 = (101*2^r-1) * (101*2^r+1)
10609*2^(2*r)-1 = (103*2^r-1) * (103*2^r+1)
11025*2^(2*r)-1 = (105*2^r-1) * (105*2^r+1)
11449*2^(2*r)-1 = (107*2^r-1) * (107*2^r+1)
11881*2^(2*r)-1 = (109*2^r-1) * (109*2^r+1)
12321*2^(2*r)-1 = (111*2^r-1) * (111*2^r+1)
12769*2^(2*r)-1 = (113*2^r-1) * (113*2^r+1)
13225*2^(2*r)-1 = (115*2^r-1) * (115*2^r+1)
13689*2^(2*r)-1 = (117*2^r-1) * (117*2^r+1)
14161*2^(2*r)-1 = (119*2^r-1) * (119*2^r+1)
14641*2^(2*r)-1 = (121*2^r-1) * (121*2^r+1)
15129*2^(2*r)-1 = (123*2^r-1) * (123*2^r+1)
15625*2^(2*r)-1 = (125*2^r-1) * (125*2^r+1)
16129*2^(2*r)-1 = (127*2^r-1) * (127*2^r+1)
16641*2^(2*r)-1 = (129*2^r-1) * (129*2^r+1)
17161*2^(2*r)-1 = (131*2^r-1) * (131*2^r+1)
17689*2^(2*r)-1 = (133*2^r-1) * (133*2^r+1)
18225*2^(2*r)-1 = (135*2^r-1) * (135*2^r+1)
18769*2^(2*r)-1 = (137*2^r-1) * (137*2^r+1)
19321*2^(2*r)-1 = (139*2^r-1) * (139*2^r+1)
19881*2^(2*r)-1 = (141*2^r-1) * (141*2^r+1)
Period 3:
27*2^(3*r)-1 = (3*2^r-1) * (9*2^(2*r)+3*2^r+1)
27*2^(3*r)+1 = (3*2^r+1) * (9*2^(2*r)-3*2^r+1)
125*2^(3*r)-1 = (5*2^r-1) * (25*2^(2*r)+5*2^r+1)
125*2^(3*r)+1 = (5*2^r+1) * (25*2^(2*r)-5*2^r+1)
343*2^(3*r)-1 = (7*2^r-1) * (49*2^(2*r)+7*2^r+1)
343*2^(3*r)+1 = (7*2^r+1) * (49*2^(2*r)-7*2^r+1)
729*2^(3*r)-1 = (9*2^r-1) * (81*2^(2*r)+9*2^r+1)
729*2^(3*r)+1 = (9*2^r+1) * (81*2^(2*r)-9*2^r+1)
1331*2^(3*r)-1 = (11*2^r-1) * (121*2^(2*r)+11*2^r+1)
1331*2^(3*r)+1 = (11*2^r+1) * (121*2^(2*r)-11*2^r+1)
2197*2^(3*r)-1 = (13*2^r-1) * (169*2^(2*r)+13*2^r+1)
2197*2^(3*r)+1 = (13*2^r+1) * (169*2^(2*r)-13*2^r+1)
3375*2^(3*r)-1 = (15*2^r-1) * (225*2^(2*r)+15*2^r+1)
3375*2^(3*r)+1 = (15*2^r+1) * (225*2^(2*r)-15*2^r+1)
4913*2^(3*r)-1 = (17*2^r-1) * (289*2^(2*r)+17*2^r+1)
4913*2^(3*r)+1 = (17*2^r+1) * (289*2^(2*r)-17*2^r+1)
6859*2^(3*r)-1 = (19*2^r-1) * (361*2^(2*r)+19*2^r+1)
6859*2^(3*r)+1 = (19*2^r+1) * (361*2^(2*r)-19*2^r+1)
9261*2^(3*r)-1 = (21*2^r-1) * (441*2^(2*r)+21*2^r+1)
9261*2^(3*r)+1 = (21*2^r+1) * (441*2^(2*r)-21*2^r+1)
12167*2^(3*r)-1 = (23*2^r-1) * (529*2^(2*r)+23*2^r+1)
12167*2^(3*r)+1 = (23*2^r+1) * (529*2^(2*r)-23*2^r+1)
15625*2^(3*r)-1 = (25*2^r-1) * (625*2^(2*r)+25*2^r+1)
15625*2^(3*r)+1 = (25*2^r+1) * (625*2^(2*r)-25*2^r+1)
19683*2^(3*r)-1 = (27*2^r-1) * (729*2^(2*r)+27*2^r+1)
19683*2^(3*r)+1 = (27*2^r+1) * (729*2^(2*r)-27*2^r+1)
Period 4:
81*2^(4*r)-1 = (3*2^r-1) * (3*2^r+1) * (9*2^(2*r)+1)
81*2^(4*r+2)+1 = (9*2^(2*r+1)-3*2^(r+1)+1) * (9*2^(2*r+1)+3*2^(r+1)+1)
625*2^(4*r)-1 = (5*2^r-1) * (5*2^r+1) * (25*2^(2*r)+1)
625*2^(4*r+2)+1 = (25*2^(2*r+1)-5*2^(r+1)+1) * (25*2^(2*r+1)+5*2^(r+1)+1)
2401*2^(4*r)-1 = (7*2^r-1) * (7*2^r+1) * (49*2^(2*r)+1)
2401*2^(4*r+2)+1 = (49*2^(2*r+1)-7*2^(r+1)+1) * (49*2^(2*r+1)+7*2^(r+1)+1)
6561*2^(4*r)-1 = (9*2^r-1) * (9*2^r+1) * (81*2^(2*r)+1)
6561*2^(4*r+2)+1 = (81*2^(2*r+1)-9*2^(r+1)+1) * (81*2^(2*r+1)+9*2^(r+1)+1)
14641*2^(4*r)-1 = (11*2^r-1) * (11*2^r+1) * (121*2^(2*r)+1)
14641*2^(4*r+2)+1 = (121*2^(2*r+1)-11*2^(r+1)+1) * (121*2^(2*r+1)+11*2^(r+1)+1)
Period 5:
243*2^(5*r)-1 = (3*2^r-1) * (81*2^(4*r)+27*2^(3*r)+9*2^(2*r)+3*2^r+1)
243*2^(5*r)+1 = (3*2^r+1) * (81*2^(4*r)-27*2^(3*r)+9*2^(2*r)-3*2^r+1)
3125*2^(5*r)-1 = (5*2^r-1) * (625*2^(4*r)+125*2^(3*r)+25*2^(2*r)+5*2^r+1)
3125*2^(5*r)+1 = (5*2^r+1) * (625*2^(4*r)-125*2^(3*r)+25*2^(2*r)-5*2^r+1)
16807*2^(5*r)-1 = (7*2^r-1) * (2401*2^(4*r)+343*2^(3*r)+49*2^(2*r)+7*2^r+1)
16807*2^(5*r)+1 = (7*2^r+1) * (2401*2^(4*r)-343*2^(3*r)+49*2^(2*r)-7*2^r+1)
Period 6:
27*2^(6*r)+1 = (3*2^(2*r)+1) * (3*2^(2*r)-3*2^r+1) * (3*2^(2*r)+3*2^r+1)
729*2^(6*r)-1 = (3*2^r-1) * (3*2^r+1) * (9*2^(2*r)-3*2^r+1) * (9*2^(2*r)+3*2^r+1)
15625*2^(6*r)-1 = (5*2^r-1) * (5*2^r+1) * (25*2^(2*r)-5*2^r+1) * (25*2^(2*r)+5*2^r+1)
19683*2^(6*r)+1 = (27*2^(2*r)+1) * (27*2^(2*r)-9*2^r+1) * (27*2^(2*r)+9*2^r+1)
Period 7:
2187*2^(7*r)-1 = (3*2^r-1) * (729*2^(6*r)+243*2^(5*r)+81*2^(4*r)+27*2^(3*r)+9*2^(2*r)+3*2^r+1)
2187*2^(7*r)+1 = (3*2^r+1) * (729*2^(6*r)-243*2^(5*r)+81*2^(4*r)-27*2^(3*r)+9*2^(2*r)-3*2^r+1)
Period 8:
6561*2^(8*r)-1 = (3*2^r-1) * (3*2^r+1) * (9*2^(2*r)+1) * (81*2^(4*r)+1)
6561*2^(8*r+4)-1 = (9*2^(2*r+1)-1) * (9*2^(2*r+1)+1) * (9*2^(2*r+1)-3*2^(r+1)+1) * (9*2^(2*r+1)+3*2^(r+1)+1)
Period 9:
19683*2^(9*r)-1 = (3*2^r-1) * (9*2^(2*r)+3*2^r+1) * (729*2^(6*r)+27*2^(3*r)+1)
19683*2^(9*r)+1 = (3*2^r+1) * (9*2^(2*r)-3*2^r+1) * (729*2^(6*r)-27*2^(3*r)+1)
Period 10:
3125*2^(10*r)-1 = (5*2^(2*r)-1) * (25*2^(4*r)-25*2^(3*r)+15*2^(2*r)-5*2^r+1) * (25*2^(4*r)+25*2^(3*r)+15*2^(2*r)+5*2^r+1)
Period 12:
729*2^(12*r)-1 = (3*2^(2*r)-1) * (3*2^(2*r)+1) * (3*2^(2*r)-3*2^r+1) * (3*2^(2*r)+3*2^r+1) * (9*2^(4*r)+3*2^(2*r)+1)
729*2^(12*r+6)+1 = (9*2^(4*r+2)+1) * (9*2^(4*r+2)-9*2^(3*r+2)+9*2^(2*r+1)-3*2^(r+1)+1) * (9*2^(4*r+2)+9*2^(3*r+2)+9*2^(2*r+1)+3*2^(r+1)+1)
Period 18:
19683*2^(18*r)+1 = (3*2^(2*r)+1) * (3*2^(2*r)-3*2^r+1) * (3*2^(2*r)+3*2^r+1) * (27*2^(6*r)-9*2^(3*r)+1) * (27*2^(6*r)+9*2^(3*r)+1)
Small prime factors or prime-power factors:
Period 2:
3
Period 3:
7
Period 4:
5
Period 5:
31
Period 6:
9 = 3^2
Period 7:
127
Period 8:
17
Period 9:
73
Period 10:
11
Period 11:
23
89
Period 12:
13
Period 13:
8191
Period 14:
43
Period 15:
151
Period 16:
257
Period 17:
131071
Period 18:
19
27 = 3^3
Period 19:
524287
Period 20:
25 = 5^2
41
Period 21:
49 = 7^2
337
Period 22:
683
Period 23:
47
178481
Period 24:
241
Period 25:
601
1801
Period 26:
2731
Period 27:
262657
Period 28:
29
113
Period 29:
233
1103
2089
Period 30:
331
Period 31:
2147483647
Period 32:
65537
Period 33:
599479
Period 34:
43691
Period 35:
71
122921
Period 36:
37
109
Period 37:
223
616318177
Period 38:
174763
Period 39:
79
121369
Period 40:
61681
Period 41:
13367
164511353
Period 42:
5419
Period 43:
431
9719
2099863
Period 44:
397
2113
Period 45:
631
23311
Period 46:
2796203
Period 47:
2351
4513
13264529
Period 48:
97
673
Period 49:
4432676798593
Period 50:
251
4051
Period 51:
103
2143
11119
Period 52:
53
157
1613
Period 53:
6361
69431
20394401
Period 54:
81 = 3^4
87211
Period 55:
881
3191
201961
Period 56:
15790321
Period 57:
32377
1212847
Period 58:
59
3033169
Period 59:
179951
3203431780337
Period 60:
61
1321
Period 61:
2305843009213693951
Period 62:
715827883
Period 63:
92737
649657
Period 64:
641
6700417
Period 65:
145295143558111
Period 66:
67
20857
Period 67:
193707721
761838257287
Period 68:
137
953
26317
Period 69:
10052678938039
Period 70:
281
86171
Period 71:
228479
48544121
212885833
Period 72:
433
38737
Period 73:
439
2298041
9361973132609
Period 74:
1777
25781083
Period 75:
100801
10567201
Period 76:
229
457
525313
Period 77:
581283643249112959
Period 78:
22366891
Period 79:
2687
202029703
1113491139767
Period 80:
4278255361
Period 81:
2593
71119
97685839
Period 82:
83
8831418697
Period 83:
167
57912614113275649087721
Period 84:
1429
14449
Period 85:
9520972806333758431
Period 86:
2932031007403
Period 87:
4177
9857737155463
Period 88:
353
2931542417
Period 89:
618970019642690137449562111
Period 90:
18837001
Period 91:
911
112901153
23140471537
Period 92:
277
1013
1657
30269
Period 93:
658812288653553079
Period 94:
283
165768537521
Period 95:
191
420778751
30327152671
Period 96:
193
22253377
Period 97:
11447
13842607235828485645766393
Period 98:
4363953127297
Period 99:
199
153649
33057806959
Period 100:
101
125 = 5^3
8101
268501
Period 101:
7432339208719
341117531003194129
Period 102:
307
2857
6529
Period 103:
2550183799
3976656429941438590393
Period 104:
858001
308761441
Period 105:
29191
106681
152041
Period 106:
107
28059810762433
Period 107:
162259276829213363391578010288127
Period 108:
246241
279073
Period 109:
745988807
870035986098720987332873
Period 110:
121 = 11^2
2971
48912491
Period 111:
321679
26295457
319020217
Period 112:
5153
54410972897
Period 113:
3391
23279
65993
1868569
1066818132868207
Period 114:
571
160465489
Period 115:
14951
4036961
2646507710984041
Period 116:
107367629
536903681
Period 117:
937
6553
86113
7830118297
Period 118:
2833
37171
1824726041
Period 119:
239
20231
62983048367
131105292137
Period 120:
4562284561
Period 121:
727
1786393878363164227858270210279
Period 122:
768614336404564651
Period 123:
3887047
177722253954175633
Period 124:
5581
8681
49477
384773
Period 125:
269089806001
4710883168879506001
Period 126:
77158673929
Period 127:
170141183460469231731687303715884105727
Period 128:
274177
67280421310721
Period 129:
11053036065049294753459639
Period 130:
131
409891
7623851
Period 131:
263
10350794431055162386718619237468234569
Period 132:
312709
4327489
Period 133:
163537220852725398851434325720959
Period 134:
7327657
6713103182899
Period 135:
271
348031
49971617830801
Period 136:
289 = 17^2
354689
2879347902817
Period 137:
32032215596496435569
5439042183600204290159
Period 138:
139
168749965921
Period 139:
5625767248687
123876132205208335762278423601
Period 140:
7416361
47392381
Period 141:
4375578271
646675035253258729
Period 142:
56409643
13952598148481
Period 143:
724153
158822951431
5782172113400990737
Period 144:
577
487824887233
Period 145:
2679895157783862814690027494144991
Period 146:
1753
1795918038741070627
Period 147:
343 = 7^3
2741672362528725535068727
Period 148:
149
593
184481113
231769777
Period 149:
86656268566282183151
8235109336690846723986161
Period 150:
1133836730401
Period 151:
18121
55871
165799
2332951
7289088383388253664437433
Period 152:
1217
148961
24517014940753
Period 153:
919
75582488424179347083438319
Period 154:
617
78233
35532364099
Period 155:
311
961 = 31^2
11471
73471
4649919401
18158209813151
Period 156:
169 = 13^2
313
1249
3121
21841
Period 157:
852133201
60726444167
1654058017289
2134387368610417
Period 158:
201487636602438195784363
Period 159:
6679
13960201
540701761
229890275929
Period 160:
414721
44479210368001
Period 161:
1289
3188767
45076044553
14808607715315782481
Period 162:
163
243 = 3^5
135433
272010961
Period 163:
150287
704161
110211473
27669118297
36230454570129675721
Period 164:
10169
181549
12112549
43249589
Period 165:
2048568835297380486760231
Period 166:
499
1163
2657
155377
13455809771
Period 167:
2349023
79638304766856507377778616296087448490695649
Period 168:
3361
88959882481
Period 169:
4057
6740339310641
3340762283952395329506327023033
Period 170:
26831423036065352611
Period 171:
93507247
3042645634792541312037847
Period 172:
173
101653
500177
1759217765581
Period 173:
730753
1505447
70084436712553223
155285743288572277679887
Period 174:
96076791871613611
Period 175:
39551
60816001
535347624791488552837151
Period 176:
229153
119782433
43872038849
Period 177:
184081
27989941729
9213624084535989031
Period 178:
179
62020897
18584774046020617
Period 179:
359
1433
1489459109360039866456940197095433721664951999121
Period 180:
181
54001
29247661
Period 181:
43441
1164193
7648337
7923871097285295625344647665764672671
Period 182:
224771
1210483
25829691707
Period 183:
367
55633
37201708625305146303973352041
Period 184:
291280009243618888211558641
Period 185:
1587855697992791
7248808599285760001152755641
Period 186:
529510939
2903110321
Period 187:
707983
1032670816743843860998850056278950666491537
Period 188:
3761
7484047069
140737471578113
Period 189:
1560007
207617485544258392970753527
Period 190:
2281
3011347479614249131
Period 191:
383
7068569257
39940132241
332584516519201
87274497124602996457
Period 192:
18446744069414584321
Period 193:
13821503
61654440233248340616559
14732265321145317331353282383
Period 194:
971
1553
31817
1100876018364883721
Period 195:
134304196845099262572814573351
Period 196:
197
19707683773
4981857697937
Period 197:
7487
26828803997912886929710867041891989490486893845712448833
Period 198:
5347
242099935645987
Period 199:
164504919713
4884164093883941177660049098586324302977543600799
Period 200:
401
340801
2787601
3173389601