mersenneforum.org Smallest k>1 such that Phi_n(k) is prime
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2018-05-03, 21:15   #1
sweety439

Nov 2016

46458 Posts
Smallest k>1 such that Phi_n(k) is prime

I found the smallest k>1 such that Phi_n(k) is (probable) prime (where Phi is the cyclotomic polynomial) for all 1<=n<=2500, see the text file. The k has been searched for special value of n's, see these OEIS sequences.

A066180 (for prime n)
A103795 (for n=2*p with p odd prime)
A056993 (for n=2^k with k>=1)
A153438 (for n=3^k with k>=2)
A246120 (for n=2*3^k with k>=1)
A246119 (for n=3*2^k with k>=1)
A298206 (for n=9*2^k with k>=1)
A246121 (for n=6^k with k>=1)
A206418 (for n=5^k with k>=2)
A205506 (for n=6*2^i*3^j with i,j>=0)
A181980 (for n=10*2^i*5^j with i,j>=0)

Let a(n) be the smallest k>1 such that Phi_n(k) is prime, I found a(n) for all 1<=n<=2500, and according to these sequences, a(2^n) is known for all 0<=n<=21, a(3^n) is known for all 0<=n<=11, a(2*3^n) is known for all 0<=n<=10, etc. and the k's for some large n are a(2^21)=919444, a(3^12)=94259, a(2*3^11)=9087, etc. However, it seems that there is no project for finding a(n) for general n. (this a(n) is the OEIS sequence A085398)

I have already update all known a(n) (including all 1<=n<=2500 and all a(n) given by these OEIS sequences) in tye wiki page http://www.mersennewiki.org/index.ph...8k%29_is_prime.
Attached Files
 least k such that phi(n,k) is prime.txt (22.0 KB, 131 views)

Last fiddled with by sweety439 on 2018-05-03 at 21:46

 2018-05-05, 23:22 #2 JeppeSN     "Jeppe" Jan 2016 Denmark 2428 Posts You could use the text file (with commas replaced by spaces) as a new (extended) so-called b-file for that OEIS entry, A085398. /JeppeSN
2018-05-06, 01:26   #3
sweety439

Nov 2016

3×823 Posts

Quote:
 Originally Posted by JeppeSN You could use the text file (with commas replaced by spaces) as a new (extended) so-called b-file for that OEIS entry, A085398. /JeppeSN
= =I did not create OEIS b-file for A085398...

This is just a text file.

Last fiddled with by sweety439 on 2018-05-06 at 01:27

 2019-04-20, 03:59 #4 sweety439     Nov 2016 246910 Posts https://docs.google.com/document/d/1...iS_1FX5c1k/pub Prime values of cyclotomic polynomial. Last fiddled with by sweety439 on 2019-04-20 at 04:07
2019-07-04, 10:47   #5
sweety439

Nov 2016

3·823 Posts

Quote:
 Originally Posted by sweety439 I found the smallest k>1 such that Phi_n(k) is (probable) prime (where Phi is the cyclotomic polynomial) for all 1<=n<=2500, see the text file. The k has been searched for special value of n's, see these OEIS sequences. A066180 (for prime n) A103795 (for n=2*p with p odd prime) A056993 (for n=2^k with k>=1) A153438 (for n=3^k with k>=2) A246120 (for n=2*3^k with k>=1) A246119 (for n=3*2^k with k>=1) A298206 (for n=9*2^k with k>=1) A246121 (for n=6^k with k>=1) A206418 (for n=5^k with k>=2) A205506 (for n=6*2^i*3^j with i,j>=0) A181980 (for n=10*2^i*5^j with i,j>=0) Let a(n) be the smallest k>1 such that Phi_n(k) is prime, I found a(n) for all 1<=n<=2500, and according to these sequences, a(2^n) is known for all 0<=n<=21, a(3^n) is known for all 0<=n<=11, a(2*3^n) is known for all 0<=n<=10, etc. and the k's for some large n are a(2^21)=919444, a(3^12)=94259, a(2*3^11)=9087, etc. However, it seems that there is no project for finding a(n) for general n. (this a(n) is the OEIS sequence A085398) I have already update all known a(n) (including all 1<=n<=2500 and all a(n) given by these OEIS sequences) in tye wiki page http://www.mersennewiki.org/index.ph...8k%29_is_prime.
Define "Satan number" as the numbers n such that the smallest such k is > phi(n), where phi is the Euler's totient function.

The first few Satan numbers are 1, 2, 25, 37, 44, 68, 75, 82, 99, 115, 119, 125, 128, 159, 162, 179, 183, 188, 203, 213, 216, 229, 233, 243, 277, 289, 292, ...

Last fiddled with by sweety439 on 2019-07-04 at 10:51

2019-07-04, 10:51   #6
sweety439

Nov 2016

3×823 Posts

Quote:
 Originally Posted by sweety439 Define "Satan number" as the numbers n such that the smallest such k is > phi(n), where phi is the Euler's totient function. The first few Satan numbers are 1, 2, 25, 37, 44, 68, 75, 82, 99, 115, 119, 125, 128, 159, 162, 179, 183, 188, 203, 213, 216, 229, 233, 243, 277, 289, 292, ...
This definition is because the order of the n-th cyclotomic polynomial is phi(n).

For the n's such that Phi_n(phi(n)) is (probable) prime, see OEIS A070525, and it is conjectured that 3, 4, 6 and 18 are the only numbers n such that the smallest such k is exactly phi(n)

Last fiddled with by sweety439 on 2019-07-04 at 10:51

2019-07-04, 10:57   #7
sweety439

Nov 2016

3×823 Posts

Quote:
 Originally Posted by sweety439 This definition is because the order of the n-th cyclotomic polynomial is phi(n). For the n's such that Phi_n(phi(n)) is (probable) prime, see OEIS A070525, and it is conjectured that 3, 4, 6 and 18 are the only numbers n such that the smallest such k is exactly phi(n)
There are the Satan numbers up to 2500 and their corresponding k's and their (Euler's) totients.
Attached Files
 Satan numbers up to 2500.txt (3.8 KB, 100 views)

 2019-07-04, 11:06 #8 sweety439     Nov 2016 3×823 Posts Also, the "satanic" of a number n is defined as.... (the smallest such k) / (phi(n)) where phi is the Euler's totient function. A number n is a Satan number if and only if its satanic is >1 If there are no such k for a number n, then the satanic of n is ∞ (however, it is conjectured such k exists for all numbers n) It is conjectured that * There are infinitely many Satan numbers. * Almost all numbers are not Satan numbers. * The satanic of a number has no upper bound. * The satanic of a random number is about 0.4
2019-10-03, 02:25   #9
sweety439

Nov 2016

3·823 Posts

Update the text file of the numbers k>1 such that Phi(n,k) is prime, for fixed n.
Attached Files
 numbers k such that cyclotomic is prime.txt (162.3 KB, 103 views)

2020-06-03, 18:11   #10
sweety439

Nov 2016

3·823 Posts

Update text file.

Code:
Smallest k>=2 such that Phi_n(k) is prime:

* n<=2500:

n

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1016

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332

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603

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190

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97

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1710

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38

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170

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33

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294

686

516

217

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542

79

210

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497

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112

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16

18

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392

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1852

11

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66

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285

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1330

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45

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1509

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313

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863

19

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63

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1350+

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717

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298

289

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677

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532

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175

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378

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234

41

2210

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32

2

511

862

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24

675

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70

136

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901

191

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1400+

342

2

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734

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168

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1425+

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364

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54

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3

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153

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602

849

401

41

527

1450+

5047

10

403

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128

501

372

1399

7

208

3

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2

179

18

86

6

2906

11

558

678

558

20

31

1475+

876

365

123

40

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6288

47

446

208

697

964

20

2707

603

17

787

1547

34

25

2047

153

338

29

358

84

1500+

614

344

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270

691

1123

65

374

707

1319

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461

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1492

450

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22

1525+

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15

143

299

6

14

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825

34

833

63

988

70

227

9

1314

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223

1550+

480

120

1966

2

24

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966

360

1410

88

1054

2

271

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4994

233

1203

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343

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22

140

439

382

1575+

765

1446

22

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515

52

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1232

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653

3

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1600+

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1962

391

182

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154

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277

6

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1625+

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234

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1080

124

307

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34

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147

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29

1650+

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2

182

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282

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1675+

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814

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1725+

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3186

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10

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40

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400

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29

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1775+

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387

2

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485

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1001

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30

209

309

413

393

8

393

311

462

127

146

139

1800+

316

325

424

287

877

143

5

43

88

80

3615

10

107

2631

41

97

681

228

613

39

330

511

2749

346

585

1825+

385

2070

26

1233

465

199

740

56

2

37

10

107

13

619

794

989

199

514

10

336

1206

6298

225

306

161

1850+

137

257

1464

114

829

103

1831

520

4034

272

316

10

3

65

83

239

381

240

200

62

1717

91

791

60

76

1875+

355

506

126

2614

451

536

18

1458

239

886

668

507

395

250

201

1697

525

567

394

260

148

116

16

142

63

1900+

687

20

1772

2

1006

1759

11

88

517

1135

125

16

180

221

2762

719

78

179

384

18

395

439

1055

6

14

1925+

35

3057

544

811

596

4223

561

467

17

1172

76

79

587

474

762

264

84

21

25

76

251

341

592

78

394

1950+

749

2189

124

1321

181

174

253

188

7

110

410

58

1040

359

91

107

402

431

195

444

12

16

6162

113

275

1975+

195

300

943

2161

192

1336

54

565

545

3

15

7631

22

139

2

389

2

375

223

151

1440

977

18

838

521

2000+

274

829

508

94

521

633

821

2

200

45

259

63

156

773

383

352

2271

9

260

47

1612

24

50

1208

157

2025+

1808

1552

2661

11

480

1716

13

146

209

608

515

2

1801

1432

8

4118

1051

226

612

233

237

304

824

180

25

2050+

76

402

249

124

634

908

180

149

1433

24

608

28

476

587

199

81

7

1077

3748

59

2216

277

119

1447

240

2075+

133

505

990

2513

605

1715

240

56

89

843

204

155

12

297

231

50

291

15

5

592

6

506

196

337

171

2100+

938

670

978

48

215

17

1185

557

714

116

85

1013

5897

980

56

70

1066

67

45

1141

175

31

237

305

409

2125+

38

383

1959

1406

291

2120

3

275

10

170

11

322

206

1088

872

1727

22

677

60

1109

13

1384

173

385

689

2150+

1279

121

1646

67

644

246

40

248

30

122

1721

1341

709

836

876

419

3709

1733

2508

229

33

531

211

84

134

2175+

6

38

229

332

23

945

15

1180

34

31

815

260

3

398

336

2412

236

699

18

567

3

2459

65

416

118

2200+

825

15

2

30

357

89

1094

24

1153

118

1717

62

447

166

1215

485

38

164

654

352

498

45

3068

1070

30

2225+

228

388

275

412

326

123

10

300

229

350

1246

410

55

403

4

41

794

119

147

38

425

520

36

75

74

2250+

520

177

1381

981

416

96

276

55

437

7

764

478

92

10

210

158

55

33

737

44

335

18

2332

133

594

2275+

47

34

372

95

273

2

197

3812

51

274

1492

955

11

403

422

875

920

2753

1398

111

171

992

2

542

108

2300+

2757

13

63

569

1919

50

136

546

1929

81

1771

959

1261

362

752

308

57

101

200

51

2718

365

3602

460

35

2325+

20

2295

142

177

109

294

14

58

2057

390

444

342

462

5367

3

115

134

21

780

61

213

1078

23

1430

350

2350+

1602

182

2

40

84

19

2538

49

4663

104

901

1133

380

353

87

424

181

12

5

411

69

144

55

105

2013

2375+

179

1605

78

2539

7

2346

672

3567

556

543

352

112

431

56

220

17

263

1048

127

228

155

602

307

3984

255

2400+

40

355

845

959

1037

2

72

115

6

1104

3197

52

568

58

60

651

412

8

981

59

464

1266

407

1385

359

2425+

1511

942

1003

1221

38

1217

48

483

40

2365

47

46

308

468

320

1442

77

1267

433

1275

658

645

202

472

256

2450+

238

175

2566

49

669

2

521

14

3130

3

1266

351

181

6

3933

220

627

34

1514

322

483

175

284

329

93

2475+

712

108

224

1893

1219

234

12

902

376

99

303

31

348

562

32

384

1444

799

59

1240

1291

1091

25

2

911

* some n>2500: (all are given by OEIS sequences)

n

smallest k>1 such that Phi_n(k) is (probable) prime

2560 = 2^9*5

1156

2592 = 2^5*3^4

646

2916 = 2^2*3^6

141

3072 = 2^10*3

129

3125 = 5^5

1527

3200 = 2^7*5^2

1619

3456 = 2^7*3^3

278

3888 = 2^4*3^5

5

4000 = 2^5*5^3

647

4096 = 2^12

150

4374 = 2*3^7

421

4608 = 2^9*3^2

224

5000 = 2^3*5^4

511

5120 = 2^10*5

34

5184 = 2^6*3^4

629

5832 = 2^3*3^6

26

6144 = 2^11*3

424

6250 = 2*5^5

2336

6400 = 2^8*5^2

2123

6561 = 3^8

17

6912 = 2^8*3^3

1081

7776 = 2^5*3^5

688

8000 = 2^6*5^3

1274

8192 = 2^13

1534

8748 = 2^2*3^7

246

9216 = 2^10*3^2

736

10000 = 2^4*5^4

2866

10240 = 2^11*5

951

10368 = 2^7*3^4

4392

11664 = 2^4*3^6

124

12288 = 2^12*3

484

12500 = 2^2*5^5

2199

12800 = 2^9*5^2

1353

13122 = 2*3^8

759

13824 = 2^9*3^3

791

15552 = 2^6*3^5

4401

15625 = 5^6

18453

16000 = 2^7*5^3

4965

16384 = 2^14

30406

17496 = 2^3*3^7

863

18432 = 2^11*3^2

2854

19683 = 3^9

3311

20000 = 2^5*5^4

7396

20480 = 2^12*5

13513

20736 = 2^8*3^4

410

23328 = 2^5*3^6

1044

24576 = 2^13*3

22

25000 = 2^3*5^5

3692

25600 = 2^10*5^2

14103

26244 = 2^2*3^8

848

27648 = 2^10*3^3

1402

31104 = 2^7*3^5

2006

31250 = 2*5^6

32275

32000 = 2^8*5^3

2257

32768 = 2^15

67234

36864 = 2^12*3^2

21234

39366 = 2*3^9

7426

40000 = 2^6*5^4

86

40960 = 2^13*5

3928

46656 = 2^6*3^6

7003

49152 = 2^14*3

5164

50000 = 2^4*5^5

2779

51200 = 2^11*5^2

18781

59049 = 3^10

4469

62500 = 2^2*5^6

85835

64000 = 2^9*5^3

820

65536 = 2^16

70906

73728 = 2^13*3^2

14837

78125 = 5^7

5517

80000 = 2^7*5^4

16647

81920 = 2^14*5

2468

98304 = 2^15*3

7726

100000 = 2^5*5^5

26677

102400 = 2^12*5^2

1172

118098 = 2*3^10

9087

125000 = 2^3*5^6

38361

128000 = 2^10*5^3

40842

131072 = 2^17

48594

147456 = 2^14*3^2

165394

177147 = 3^11

94259

196608 = 2^16*3

13325

262144 = 2^18

62722

279936 = 2^7*3^7

1925

294912 = 2^15*3^2

24743

393216 = 2^17*3

96873

524288 = 2^19

24518

589824 = 2^16*3^2

62721

786432 = 2^18*3

192098

1048576 = 2^20

75898

1179648 = 2^17*3^2

237804

2097152 = 2^21

919444

2359296 = 2^18*3^2

143332

* some unknown terms: (all are given by OEIS sequences)

n

smallest k>1 such that Phi_n(k) is (probable) prime

354294 = 2*3^11

>35000

1062882 = 2*3^12

>3500

1572864 = 2^19*3

<=712012

3145728 = 2^20*3

<=123447

4194304 = 2^22

>195000

8388608 = 2^23

>109000

Let a(n) be the smallest k>=2 such that Phi_n(k) is prime.

It is conjectured that

* a(n) exists for every n>=1

* a(n) ~ phi(n)*gamma, where phi is the Euler totient function, and gamma is the Euler-Mascheroni constant (0.577215664901…)

OEIS sequences:

a(n): A085398

a(n) for prime n: A066180

a(2*n) for odd prime n: A103795

a(2^n) for n>=1: A056993

a(3^n) for n>=2: A153438

a(2*3^n) for n>=1: A246120

a(3*2^n) for n>=1: A246119

a(9*2^n) for n>=1: A298206

a(6^n) for n>=1: A246121

a(5^n) for n>=2: A206418

a(6*n) for n of the form 2^i*3^j with i,j>=0: A205506

a(10*n) for n of the form 2^i*5^j with i,j>=0: A181980
Attached Files
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 2020-10-18, 09:25 #11 sweety439     Nov 2016 46458 Posts

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