Asymptotic factorizations of an Eulerproduct into primefactors.

 

Let : : where the q are prime. We are considering limN->oo wN,s (mod pk) where the pk are primes and here pk=2...89

and estimate the relative frequency of primefactors pk after evaluating the euler-product wN,s with exponent s up to the Nth prime.

 

Tables: absolute and (scaled) relative count of occurences of primefactor pk in relation to N, formula: rel=(count/N) * (pk-1)

N=200

Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(200)

 

s\p

2

3

5

7

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

 

1

392

142

61

35

20

17

12

9

10

6

4

3

3

4

5

3

4

3

1

2

3

1

4

2

 

2

784

294

114

71

40

35

23

20

17

12

11

8

7

10

9

6

6

6

3

4

4

4

6

2

 

3

392

238

61

107

20

50

12

35

10

6

18

14

3

12

5

3

4

7

8

2

10

5

4

2

 

4

983

294

245

71

40

75

48

20

17

27

11

18

16

10

9

13

6

13

3

4

8

4

6

6

 

5

392

142

110

35

107

17

12

9

10

6

32

3

21

4

5

3

4

14

1

14

3

1

4

2

 

6

784

493

114

224

40

104

23

69

17

12

40

28

7

28

9

6

6

17

15

4

18

13

6

2

 

7

392

142

61

67

20

17

12

9

10

54

4

3

3

31

5

3

4

3

1

18

3

1

4

2

 

8

1182

294

245

71

40

75

99

20

17

27

11

18

38

10

9

13

6

13

3

4

16

4

6

15

 

9

392

334

61

107

20

50

12

105

10

6

18

52

3

12

5

3

4

7

8

2

24

5

4

2

 

10

784

294

208

71

217

35

23

20

17

12

69

8

50

10

9

6

6

33

3

28

4

4

6

2

 

11

392

142

61

35

39

17

12

9

98

6

4

3

3

4

5

3

4

3

36

2

3

1

4

24

 

12

983

493

245

224

40

216

48

69

17

27

40

61

16

28

9

13

6

37

15

4

34

13

6

6

 

 

relative frequency: count/N*(pk - 1)

 

 

1

1.96

2.84

4.88

6.30

10.00

12.24

15.36

14.58

24.20

23.52

18.00

19.44

24.00

35.28

52.90

40.56

67.28

54.00

21.78

49.00

77.76

30.42

134.48

77.44

 

2

3.92

5.88

9.12

12.78

20.00

25.20

29.44

32.40

41.14

47.04

49.50

51.84

56.00

88.20

95.22

81.12

100.92

108.00

65.34

98.00

103.68

121.68

201.72

77.44

 

3

1.96

4.76

4.88

19.26

10.00

36.00

15.36

56.70

24.20

23.52

81.00

90.72

24.00

105.84

52.90

40.56

67.28

126.00

174.24

49.00

259.20

152.10

134.48

77.44

 

4

4.92

5.88

19.60

12.78

20.00

54.00

61.44

32.40

41.14

105.84

49.50

116.64

128.00

88.20

95.22

175.76

100.92

234.00

65.34

98.00

207.36

121.68

201.72

232.32

 

5

1.96

2.84

8.80

6.30

53.50

12.24

15.36

14.58

24.20

23.52

144.00

19.44

168.00

35.28

52.90

40.56

67.28

252.00

21.78

343.00

77.76

30.42

134.48

77.44

 

6

3.92

9.86

9.12

40.32

20.00

74.88

29.44

111.78

41.14

47.04

180.00

181.44

56.00

246.96

95.22

81.12

100.92

306.00

326.70

98.00

466.56

395.46

201.72

77.44

 

7

1.96

2.84

4.88

12.06

10.00

12.24

15.36

14.58

24.20

211.68

18.00

19.44

24.00

273.42

52.90

40.56

67.28

54.00

21.78

441.00

77.76

30.42

134.48

77.44

 

8

5.91

5.88

19.60

12.78

20.00

54.00

126.72

32.40

41.14

105.84

49.50

116.64

304.00

88.20

95.22

175.76

100.92

234.00

65.34

98.00

414.72

121.68

201.72

580.80

 

9

1.96

6.68

4.88

19.26

10.00

36.00

15.36

170.10

24.20

23.52

81.00

336.96

24.00

105.84

52.90

40.56

67.28

126.00

174.24

49.00

622.08

152.10

134.48

77.44

 

10

3.92

5.88

16.64

12.78

108.50

25.20

29.44

32.40

41.14

47.04

310.50

51.84

400.00

88.20

95.22

81.12

100.92

594.00

65.34

686.00

103.68

121.68

201.72

77.44

 

11

1.96

2.84

4.88

6.30

19.50

12.24

15.36

14.58

237.16

23.52

18.00

19.44

24.00

35.28

52.90

40.56

67.28

54.00

784.08

49.00

77.76

30.42

134.48

929.28

 

12

4.92

9.86

19.60

40.32

20.00

155.52

61.44

111.78

41.14

105.84

180.00

395.28

128.00

246.96

95.22

175.76

100.92

666.00

326.70

98.00

881.28

395.46

201.72

232.32

 

 

 

113

 

N=1000

Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(1000)

 

s\p

2

3

5

7

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

 

1

1961

736

307

191

110

85

65

50

48

33

31

28

24

24

23

17

19

17

18

13

14

13

11

12

 

2

3952

1494

613

385

219

173

129

110

96

68

67

54

48

48

44

36

35

30

34

27

24

25

21

21

 

3

1961

1226

307

573

110

261

65

175

48

33

103

88

24

73

23

17

19

54

50

13

42

40

11

12

 

4

4951

1494

1253

385

219

356

265

110

96

144

67

107

97

48

44

74

35

65

34

27

49

25

21

45

 

5

1961

736

552

191

545

85

65

50

48

33

176

28

124

24

23

17

19

83

18

69

14

13

11

12

 

6

3952

2493

613

1160

219

527

129

349

96

68

205

165

48

142

44

36

35

98

95

27

79

78

21

21

 

7

1961

736

307

356

110

85

65

50

48

258

31

28

24

166

23

17

19

17

18

98

14

13

11

12

 

8

5950

1494

1253

385

219

356

523

110

96

144

67

107

200

48

44

74

35

65

34

27

103

25

21

81

 

9

1961

1716

307

573

110

261

65

513

48

33

103

262

24

73

23

17

19

54

50

13

119

40

11

12

 

10

3952

1494

1104

385

1090

173

129

110

96

68

347

54

253

48

44

36

35

166

34

138

24

25

21

21

 

11

1961

736

307

191

211

85

65

50

510

33

31

28

24

24

23

17

19

17

176

13

14

13

11

129

 

12

4951

2493

1253

1160

219

1080

265

349

96

144

205

334

97

142

44

74

35

202

95

27

163

78

21

45

 

 

relative frequency: count/N*(pk - 1)

 

 

1

1.96

2.94

4.91

6.88

11.00

12.24

16.64

16.20

23.23

25.87

27.90

36.29

38.40

42.34

48.67

45.97

63.92

61.20

78.41

63.70

72.58

79.09

73.96

92.93

 

2

3.95

5.98

9.81

13.86

21.90

24.91

33.02

35.64

46.46

53.31

60.30

69.98

76.80

84.67

93.10

97.34

117.74

108.00

148.10

132.30

124.42

152.10

141.20

162.62

 

3

1.96

4.90

4.91

20.63

11.00

37.58

16.64

56.70

23.23

25.87

92.70

114.05

38.40

128.77

48.67

45.97

63.92

194.40

217.80

63.70

217.73

243.36

73.96

92.93

 

4

4.95

5.98

20.05

13.86

21.90

51.26

67.84

35.64

46.46

112.90

60.30

138.67

155.20

84.67

93.10

200.10

117.74

234.00

148.10

132.30

254.02

152.10

141.20

348.48

 

5

1.96

2.94

8.83

6.88

54.50

12.24

16.64

16.20

23.23

25.87

158.40

36.29

198.40

42.34

48.67

45.97

63.92

298.80

78.41

338.10

72.58

79.09

73.96

92.93

 

6

3.95

9.97

9.81

41.76

21.90

75.89

33.02

113.08

46.46

53.31

184.50

213.84

76.80

250.49

93.10

97.34

117.74

352.80

413.82

132.30

409.54

474.55

141.20

162.62

 

7

1.96

2.94

4.91

12.82

11.00

12.24

16.64

16.20

23.23

202.27

27.90

36.29

38.40

292.82

48.67

45.97

63.92

61.20

78.41

480.20

72.58

79.09

73.96

92.93

 

8

5.95

5.98

20.05

13.86

21.90

51.26

133.89

35.64

46.46

112.90

60.30

138.67

320.00

84.67

93.10

200.10

117.74

234.00

148.10

132.30

533.95

152.10

141.20

627.26

 

9

1.96

6.86

4.91

20.63

11.00

37.58

16.64

166.21

23.23

25.87

92.70

339.55

38.40

128.77

48.67

45.97

63.92

194.40

217.80

63.70

616.90

243.36

73.96

92.93

 

10

3.95

5.98

17.66

13.86

109.00

24.91

33.02

35.64

46.46

53.31

312.30

69.98

404.80

84.67

93.10

97.34

117.74

597.60

148.10

676.20

124.42

152.10

141.20

162.62

 

11

1.96

2.94

4.91

6.88

21.10

12.24

16.64

16.20

246.84

25.87

27.90

36.29

38.40

42.34

48.67

45.97

63.92

61.20

766.66

63.70

72.58

79.09

73.96

998.98

 

12

4.95

9.97

20.05

41.76

21.90

155.52

67.84

113.08

46.46

112.90

184.50

432.86

155.20

250.49

93.10

200.10

117.74

727.20

413.82

132.30

844.99

474.55

141.20

348.48

 

 

 

 

N=2000

Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(2000)

 

s\p

2

3

5

7

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

 

1

3965

1479

617

384

221

183

132

107

91

75

62

55

50

47

46

34

36

35

31

29

28

27

26

21

 

2

7979

2986

1226

777

440

359

259

233

184

146

129

112

97

97

86

73

67

65

55

60

53

51

47

43

 

3

3965

2469

617

1154

221

541

132

347

91

75

205

170

50

152

46

34

36

106

90

29

85

82

26

21

 

4

9978

2986

2494

777

440

721

522

233

184

296

129

230

199

97

86

152

67

134

55

60

105

51

47

90

 

5

3965

1479

1112

384

1102

183

132

107

91

75

332

55

244

47

46

34

36

162

31

143

28

27

26

21

 

6

7979

4985

1226

2333

440

1072

259

711

184

146

406

340

97

298

86

73

67

201

174

60

168

165

47

43

 

7

3965

1479

617

716

221

183

132

107

91

505

62

55

50

338

46

34

36

35

31

199

28

27

26

21

 

8

11977

2986

2494

777

440

721

1059

233

184

296

129

230

403

97

86

152

67

134

55

60

212

51

47

179

 

9

3965

3459

617

1154

221

541

132

1034

91

75

205

503

50

152

46

34

36

106

90

29

252

82

26

21

 

10

7979

2986

2210

777

2195

359

259

233

184

146

673

112

504

97

86

73

67

330

55

284

53

51

47

43

 

11

3965

1479

617

384

420

183

132

107

1040

75

62

55

50

47

46

34

36

35

340

29

28

27

26

250

 

12

9978

4985

2494

2333

440

2166

522

711

184

296

406

685

199

298

86

152

67

408

174

60

334

165

47

90

 

 

relative frequency: count/N*(pk - 1)

 

 

1

1.98

2.96

4.94

6.91

11.05

13.18

16.90

17.33

22.02

29.40

27.90

35.64

40.00

41.45

48.67

45.97

60.55

63.00

67.52

71.05

72.58

82.13

87.41

81.31

 

2

3.99

5.97

9.81

13.99

22.00

25.85

33.15

37.75

44.53

57.23

58.05

72.58

77.60

85.55

90.99

98.70

112.69

117.00

119.79

147.00

137.38

155.14

158.01

166.50

 

3

1.98

4.94

4.94

20.77

11.05

38.95

16.90

56.21

22.02

29.40

92.25

110.16

40.00

134.06

48.67

45.97

60.55

190.80

196.02

71.05

220.32

249.44

87.41

81.31

 

4

4.99

5.97

19.95

13.99

22.00

51.91

66.82

37.75

44.53

116.03

58.05

149.04

159.20

85.55

90.99

205.50

112.69

241.20

119.79

147.00

272.16

155.14

158.01

348.48

 

5

1.98

2.96

8.90

6.91

55.10

13.18

16.90

17.33

22.02

29.40

149.40

35.64

195.20

41.45

48.67

45.97

60.55

291.60

67.52

350.35

72.58

82.13

87.41

81.31

 

6

3.99

9.97

9.81

41.99

22.00

77.18

33.15

115.18

44.53

57.23

182.70

220.32

77.60

262.84

90.99

98.70

112.69

361.80

378.97

147.00

435.46

501.93

158.01

166.50

 

7

1.98

2.96

4.94

12.89

11.05

13.18

16.90

17.33

22.02

197.96

27.90

35.64

40.00

298.12

48.67

45.97

60.55

63.00

67.52

487.55

72.58

82.13

87.41

81.31

 

8

5.99

5.97

19.95

13.99

22.00

51.91

135.55

37.75

44.53

116.03

58.05

149.04

322.40

85.55

90.99

205.50

112.69

241.20

119.79

147.00

549.50

155.14

158.01

693.09

 

9

1.98

6.92

4.94

20.77

11.05

38.95

16.90

167.51

22.02

29.40

92.25

325.94

40.00

134.06

48.67

45.97

60.55

190.80

196.02

71.05

653.18

249.44

87.41

81.31

 

10

3.99

5.97

17.68

13.99

109.75

25.85

33.15

37.75

44.53

57.23

302.85

72.58

403.20

85.55

90.99

98.70

112.69

594.00

119.79

695.80

137.38

155.14

158.01

166.50

 

11

1.98

2.96

4.94

6.91

21.00

13.18

16.90

17.33

251.68

29.40

27.90

35.64

40.00

41.45

48.67

45.97

60.55

63.00

740.52

71.05

72.58

82.13

87.41

968.00

 

12

4.99

9.97

19.95

41.99

22.00

155.95

66.82

115.18

44.53

116.03

182.70

443.88

159.20

262.84

90.99

205.50

112.69

734.40

378.97

147.00

865.73

501.93

158.01

348.48

 

 

 

 

N=10000

Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(10000)

 

s\p

2

3

5

7

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

 

1

19954

7498

3110

1922

1086

898

660

574

464

369

348

280

257

251

219

193

179

172

150

153

138

134

115

106

 

2

39960

14984

6211

3857

2202

1789

1322

1169

941

738

688

567

515

489

444

385

347

346

294

293

279

257

238

221

 

3

19954

12486

3110

5783

1086

2702

660

1750

464

369

1047

863

257

739

219

193

179

516

458

153

423

390

115

106

 

4

49959

14984

12495

3857

2202

3586

2640

1169

941

1489

688

1137

1024

489

444

784

347

678

294

293

561

257

238

442

 

5

19954

7498

5594

1922

5487

898

660

574

464

369

1737

280

1257

251

219

193

179

835

150

737

138

134

115

106

 

6

39960

24983

6211

11642

2202

5408

1322

3520

941

738

2080

1714

515

1443

444

385

347

1030

915

293

848

774

238

221

 

7

19954

7498

3110

3583

1086

898

660

574

464

2566

348

280

257

1712

219

193

179

172

150

1006

138

134

115

106

 

8

59958

14984

12495

3857

2202

3586

5297

1169

941

1489

688

1137

2045

489

444

784

347

678

294

293

1130

257

238

899

 

9

19954

17474

3110

5783

1086

2702

660

5274

464

369

1047

2554

257

739

219

193

179

516

458

153

1268

390

115

106

 

10

39960

14984

11186

3857

11007

1789

1322

1169

941

738

3443

567

2561

489

444

385

347

1689

294

1442

279

257

238

221

 

11

19954

7498

3110

1922

2076

898

660

574

5204

369

348

280

257

251

219

193

179

172

1680

153

138

134

115

1245

 

12

49959

24983

12495

11642

2202

10840

2640

3520

941

1489

2080

3434

1024

1443

444

784

347

2041

915

293

1700

774

238

442

 

 

relative frequency rounded to integer: count/N*(pk - 1)

 

 

1

2

3

5

7

11

13

17

19

22

29

31

36

41

44

46

52

60

62

65

75

72

82

77

82

 

2

4

6

10

14

22

26

34

38

46

58

62

73

82

86

94

104

117

125

128

144

145

156

160

171

 

3

2

5

5

21

11

39

17

57

22

29

94

112

41

130

46

52

60

186

200

75

219

237

77

82

 

4

5

6

20

14

22

52

68

38

46

117

62

147

164

86

94

212

117

244

128

144

291

156

160

342

 

5

2

3

9

7

55

13

17

19

22

29

156

36

201

44

46

52

60

301

65

361

72

82

77

82

 

6

4

10

10

42

22

78

34

114

46

58

187

222

82

255

94

104

117

371

399

144

440

471

160

171

 

7

2

3

5

13

11

13

17

19

22

201

31

36

41

302

46

52

60

62

65

493

72

82

77

82

 

8

6

6

20

14

22

52

136

38

46

117

62

147

327

86

94

212

117

244

128

144

586

156

160

696

 

9

2

7

5

21

11

39

17

171

22

29

94

331

41

130

46

52

60

186

200

75

657

237

77

82

 

10

4

6

18

14

110

26

34

38

46

58

310

73

410

86

94

104

117

608

128

707

145

156

160

171

 

11

2

3

5

7

21

13

17

19

252

29

31

36

41

44

46

52

60

62

732

75

72

82

77

964

 

12

5

10

20

42

22

156

68

114

46

117

187

445

164

255

94

212

117

735

399

144

881

471

160

342

 

 

 

 

 

 

 

 

N=20000

Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(20000)

 

s\p

2

3

5

7

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

 

1

40152

15083

6247

3897

2195

1812

1332

1170

946

742

683

574

524

489

444

394

357

336

307

294

282

273

236

231

 

2

80415

30152

12538

7804

4393

3613

2665

2351

1902

1481

1380

1147

1042

984

898

786

711

691

614

578

555

528

491

475

 

3

40152

25127

6247

11690

2195

5402

1332

3552

946

742

2063

1724

524

1511

444

394

357

1012

917

294

841

779

236

231

 

4

100537

30152

25157

7804

4393

7249

5351

2351

1902

2978

1380

2289

2074

984

898

1581

711

1376

614

578

1132

528

491

929

 

5

40152

15083

11256

3897

11040

1812

1332

1170

946

742

3454

574

2544

489

444

394

357

1707

307

1459

282

273

236

231

 

6

80415

50274

12538

23461

4393

10852

2665

7069

1902

1481

4169

3440

1042

2965

898

786

711

2043

1856

578

1671

1558

491

475

 

7

40152

15083

6247

7245

2195

1812

1332

1170

946

5188

683

574

524

3419

444

394

357

336

307

2005

282

273

236

231

 

8

120659

30152

25157

7804

4393

7249

10673

2351

1902

2978

1380

2289

4142

984

898

1581

711

1376

614

578

2257

528

491

1835

 

9

40152

35171

6247

11690

2195

5402

1332

10600

946

742

2063

5128

524

1511

444

394

357

1012

917

294

2574

779

236

231

 

10

80415

30152

22571

7804

22113

3613

2665

2351

1902

1481

6912

1147

5150

984

898

786

711

3423

614

2903

555

528

491

475

 

11

40152

15083

6247

3897

4201

1812

1332

1170

10477

742

683

574

524

489

444

394

357

336

3378

294

282

273

236

2541

 

12

100537

50274

25157

23461

4393

21787

5351

7069

1902

2978

4169

6928

2074

2965

898

1581

711

4077

1856

578

3374

1558

491

929

 

 

relative frequency rounded to integer: count/N*(pk - 1)

 

 

1

2

3

5

7

11

13

17

19

23

29

31

37

42

43

47

53

60

60

67

72

73

83

79

89

 

2

4

6

10

14

22

26

34

38

46

58

62

74

83

87

95

106

120

124

134

142

144

161

165

184

 

3

2

5

5

21

11

39

17

58

23

29

93

112

42

133

47

53

60

182

200

72

218

237

79

89

p=1(mod3)

 

4

5

6

20

14

22

52

68

38

46

117

62

148

166

87

95

214

120

248

134

142

293

161

165

360

 

5

2

3

9

7

55

13

17

19

23

29

155

37

204

43

47

53

60

307

67

357

73

83

79

89

p=1(mod5)

 

6

4

10

10

42

22

78

34

115

46

58

188

223

83

262

95

106

120

368

404

142

433

474

165

184

 

7

2

3

5

13

11

13

17

19

23

203

31

37

42

302

47

53

60

60

67

491

73

83

79

89

p=1(mod7)

 

8

6

6

20

14

22

52

137

38

46

117

62

148

331

87

95

214

120

248

134

142

585

161

165

711

 

9

2

7

5

21

11

39

17

172

23

29

93

332

42

133

47

53

60

182

200

72

667

237

79

89

 

10

4

6

18

14

111

26

34

38

46

58

311

74

412

87

95

106

120

616

134

711

144

161

165

184

 

11

2

3

5

7

21

13

17

19

254

29

31

37

42

43

47

53

60

60

736

72

73

83

79

984

 

12

5

10

20

42

22

157

68

115

46

117

188

449

166

262

95

214

120

734

404

142

875

474

165

360

 

 

 

 

N=41000

Count of occurences of primefactor p in the (partial) Eulerproduct over prime(1)...prime(41000)

 

s\p

2

3

5

7

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

 

1

81841

30680

12814

7923

4478

3702

2716

2385

1936

1527

1382

1167

1061

990

919

797

724

692

633

593

561

541

497

466

 

2

163877

61441

25591

15930

9021

7381

5426

4826

3879

3016

2804

2332

2112

1980

1838

1598

1445

1401

1258

1184

1123

1058

1017

936

 

3

81841

51149

12814

23840

4478

11087

2716

7196

1936

1527

4205

3503

1061

3025

919

797

724

2095

1897

593

1700

1591

497

466

 

4

204876

61441

51242

15930

9021

14805

10898

4826

3879

6024

2804

4664

4216

1980

1838

3204

1445

2797

1258

1184

2277

1058

1017

1874

 

5

81841

30680

23065

7923

22438

3702

2716

2385

1936

1527

7021

1167

5228

990

919

797

724

3468

633

2972

561

541

497

466

 

6

163877

102440

25591

47837

9021

22149

5426

14425

3879

3016

8478

7014

2112

5992

1838

1598

1445

4175

3786

1184

3421

3171

1017

936

 

7

81841

30680

12814

14732

4478

3702

2716

2385

1936

10626

1382

1167

1061

6987

919

797

724

692

633

4139

561

541

497

466

 

8

245875

61441

51242

15930

9021

14805

21766

4826

3879

6024

2804

4664

8412

1980

1838

3204

1445

2797

1258

1184

4560

1058

1017

3738

 

9

81841

71618

12814

23840

4478

11087

2716

21571

1936

1527

4205

10482

1061

3025

919

797

724

2095

1897

593

5181

1591

497

466

 

10

163877

61441

46070

15930

45072

7381

5426

4826

3879

3016

14094

2332

10496

1980

1838

1598

1445

6934

1258

5930

1123

1058

1017

936

 

11

81841

30680

12814

7923

8572

3702

2716

2385

21357

1527

1382

1167

1061

990

919

797

724

692

6881

593

561

541

497

5145

 

12

204876

102440

51242

47837

9021

44365

10898

14425

3879

6024

8478

14052

4216

5992

1838

3204

1445

8356

3786

1184

6896

3171

1017

1874

 

 

relative frequency rounded to integer: count/N*(pk - 1)

 

 

1

2

3

5

7

11

13

17

19

23

29

30

37

41

43

47

53

59

61

67

71

71

80

82

88

 

2

4

6

10

14

22

26

34

38

46

58

62

74

82

85

95

105

119

123

134

142

142

157

167

177

 

3

2

5

5

21

11

39

17

57

23

29

92

111

41

130

47

53

59

184

202

71

215

236

82

88

p=1(mod3)

 

4

5

6

20

14

22

52

68

38

46

115

62

147

165

85

95

211

119

246

134

142

288

157

167

354

 

5

2

3

9

7

55

13

17

19

23

29

154

37

204

43

47

53

59

305

67

355

71

80

82

88

p=1(mod5)

 

6

4

10

10

42

22

78

34

114

46

58

186

222

82

258

95

105

119

367

402

142

433

471

167

177

 

7

2

3

5

13

11

13

17

19

23

203

30

37

41

301

47

53

59

61

67

495

71

80

82

88

p=1(mod7)

 

8

6

6

20

14

22

52

136

38

46

115

62

147

328

85

95

211

119

246

134

142

577

157

167

706

 

9

2

7

5

21

11

39

17

170

23

29

92

331

41

130

47

53

59

184

202

71

655

236

82

88

 

10

4

6

18

14

110

26

34

38

46

58

309

74

410

85

95

105

119

609

134

709

142

157

167

177

 

11

2

3

5

7

21

13

17

19

252

29

30

37

41

43

47

53

59

61

731

71

71

80

82

972

 

12

5

10

20

42

22

156

68

114

46

115

186

444

165

258

95

211

119

734

402

142

872

471

167

354