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)

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)

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)

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)

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)

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 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)

Small prime factors:

Period 2:

3

Period 3:

7

Period 4:

5

Period 5:

31

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

Period 19:

524287

Period 20:

41

Period 21:

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:

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, 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:

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:

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:

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, 11471, 73471, 4649919401, 18158209813151

Period 156:

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, 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