ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 ·...

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ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are illustrated by assembling members of prime families into a single graph, with arrows pointing to the next member of aliquot sequence. 1. Intro An aliquot sequence is an iterative mapping of positive integers n σ(n) - n, from n to the sum of its divisors that are less than n [5, 1, 8, 7][6, Sec. B5]. These connections are tabulated in entry A001065 of the OEIS www.oeis.org [9], and can be investigated by the database in www.factordb.com. The typical track of such a mapping ends in a prime number, and this prime number is followed by a 1, which is generally not mentioned explicitly. The length of the track is entry A098007 [9]. In other cases, the track may enter a cycle, see A098007 in the OEIS [9], even a cycle of length one. In any case, we may build families of the integers having common parts of their tracks (i.e., having the same termination, A115350 in the OEIS [9]). The graphs in section 2 show some of these families, whereas section 3 considers the map n σ * (n). 2. Aliquot sequence Illustrations The following plots up to Figure 72 show some of these mergers. Where they end in a known prime, that prime is marked with a p. Obviously, each of the plots is only a part of a tree-type structure. Because I generated them by tracing a finite set of initial values, the “early” plots may appear to have more members, but this is an illusion generated from this statistical sampling artifact. In addition, I have not traced the sequences completely in all cases; this means the individual families are not complete up to the maximum number that appears in the individual plot! The plots are created with the dot command of www.graphviz.org. Date : March 21, 2014. Key words and phrases. Aliquot Sequence, Divisors. 1

Transcript of ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 ·...

Page 1: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS

RICHARD J. MATHAR

Abstract. Aliquot sequences are illustrated by assembling members of prime

families into a single graph, with arrows pointing to the next member of aliquot

sequence.

1. Intro

An aliquot sequence is an iterative mapping of positive integers n → σ(n) − n,from n to the sum of its divisors that are less than n [5, 1, 8, 7][6, Sec. B5]. Theseconnections are tabulated in entry A001065 of the OEIS www.oeis.org [9], and canbe investigated by the database in www.factordb.com.

The typical track of such a mapping ends in a prime number, and this primenumber is followed by a 1, which is generally not mentioned explicitly. The lengthof the track is entry A098007 [9]. In other cases, the track may enter a cycle, seeA098007 in the OEIS [9], even a cycle of length one. In any case, we may buildfamilies of the integers having common parts of their tracks (i.e., having the sametermination, A115350 in the OEIS [9]).

The graphs in section 2 show some of these families, whereas section 3 considersthe map n→ σ∗(n).

2. Aliquot sequence Illustrations

The following plots up to Figure 72 show some of these mergers. Where theyend in a known prime, that prime is marked with a p.

Obviously, each of the plots is only a part of a tree-type structure. BecauseI generated them by tracing a finite set of initial values, the “early” plots mayappear to have more members, but this is an illusion generated from this statisticalsampling artifact. In addition, I have not traced the sequences completely in allcases; this means the individual families are not complete up to the maximumnumber that appears in the individual plot!

The plots are created with the dot command of www.graphviz.org.

Date: March 21, 2014.Key words and phrases. Aliquot Sequence, Divisors.

1

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Figure 1. Prime family 3 [9, A127163]

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ALIQUOT SEQUENCE MERGERS 3

Figure 2. Aliquot cycle 6.

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Figure 3. Prime family 7 [9, A127164].

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Figure 5. Prime family 13

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Figure 6. Prime family 17

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Figure 9. Prime family 31

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Figure 10. Prime family 37

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61

4

73

06

64

67

0

10

76

69

2

69

8

72

1

76

4

74

2

74

8

75

3

12

76

12

44

76

8

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48

24

56

79

2

80

1

80

2

81

4

81

7

82

6

84

4

83

6

83

8

84

8

11

06

91

0

95

0

95

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96

4

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10

10

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10

66

10

71

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98

10

88

10

89

10

90

11

02

11

08

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96

13

54

11

16

11

18

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11

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11

78

11

86

26

44

19

90

12

00

12

03

17

44

16

66

12

32

12

61

13

10

13

34

13

12

13

22

13

28

21

26

13

32

13

42

13

77

13

78

14

12

14

54

14

68

14

78

14

96

15

09

15

22

15

45

21

64

15

51

15

65

15

82

15

88

15

99

18

46

16

10

16

22

16

30

16

46

22

52

16

96

16

68

16

70

17

06 22

06

17

00

22

40

38

56

17

20

17

73

17

74

18

14

18

15

18

26

18

28

18

34

18

64

18

74

18

94

19

48

19

65

20

56

21

10

21

19

21

74

21

75

21

77

21

91

22

09

36

46

22

58

22

66

22

70

22

84

22

94

23

02

24

58

23

68

23

84

23

91

24

21

24

28

24

49

24

52

24

76

25

04

25

06

25

34

25

88

26

08

26

14

41

06

26

28

26

38

26

39

26

81

26

98

27

01

27

32

27

43

27

74

27

79

27

88

27

92

27

94

28

04

28

10

28

18

28

29

28

41

44

36

33

34

28

44

31

78

28

70

28

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28

81

28

83

29

02

29

07

29

32

31

76

29

44

29

83

29

85

29

86

33

26

30

10

30

26

30

51

30

82

31

10

31

33

31

48

31

52

31

58

31

82

32

90

36

22

31

90

31

95

32

14

32

38

32

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32

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32

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44

60

49

48

33

24

37

16

33

40

33

77

33

86

33

94

40

24

35

36

34

16

34

47

34

70

34

82

34

90

34

93

35

08

35

14

42

76

35

72

35

85

35

92

36

14

36

20

36

34

36

56

36

62

37

03

37

18

37

22

37

42

37

48

38

06

42

44

38

20

38

30

38

78

39

26

39

27

39

39

39

45

39

50

39

74

39

91

39

92

40

39

40

42

40

55

40

67

40

90

41

19

45

10

45

62

41

30

41

31

41

42

41

50

41

68

41

69

41

86

42

10

42

33

46

58

42

70

43

22

43

30

43

34

43

46

43

70

43

71

44

06

44

32

44

74

44

98

45

16

45

28

45

34

46

30

46

34

46

46

46

95

47

18

47

61

47

62

48

28

48

52

48

62

48

69

48

86

49

06

49

36

49

46

49

58

49

59

49

82

49

94

Figure 11. Prime family 41

Page 13: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 13

50

43

p

40

44

10

8

17

2

60

74

70

81

82

94

13

6

13

4

11

2

11

6

20

4

30

0

13

2

14

21

47

15

3

15

4

15

8

21

8

16

0

17

01

82

18

5

19

0

31

6

24

4

19

2

56

8

22

6

20

82

31

23

2

24

8

25

0

32

8

30

2

26

0

26

22

68

28

6

27

2

27

8

29

0

29

25

12

30

4

31

4

44

2

32

0

33

4

65

6

64

6

33

6

34

1

34

4

34

5

49

2

68

4

34

8

35

8

37

6

36

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37

7

46

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54

8

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2

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2

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43

6

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1

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0

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4

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44

8

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80

6 53

8

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8

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24

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48

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48

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6

49

0

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36

49

55

06

51

1

51

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71

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3

59

0

56

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92

8

96

2

56

0

60

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57

2

57

46

10

62

2

11

12

98

8

62

4

62

6

63

2

63

4

63

8

66

2

66

3

27

20

68

1

10

96

68

6

70

5

97

2

15

76

70

8

71

2

10

04

76

0

73

2

73

4

74

6

74

7

75

4

10

40

15

64

95

8

77

0

78

2

79

68

18

82

5

94

6

83

2

11

64

15

80

85

2

85

4

85

5

86

0

86

28

66

87

1

87

4

87

8

88

6

89

5

89

7

90

2

90

4

91

4

12

40

16

40

92

0

12

92

12

28

94

8

95

6

96

3

13

94

97

3

97

4

97

69

86

99

4

10

06

10

11

10

12

10

16

10

18

10

22

14

60

10

52

10

65

10

70

10

78

10

82

10

83

10

94

10

95

11

14

11

19

11

26

11

38

11

42

11

49

11

50

11

55

11

59

11

62

17

80

11

68

11

72

11

74

12

02

12

14

12

34

12

38

12

39

21

40

12

46

12

52

12

56

12

62

12

68

17

86

12

80

12

86

12

87

12

98

12

99

17

38

13

00

13

06

13

18

13

29

15

50

14

26

13

30

13

35

40

84

13

48

13

53

13

58

13

66

13

70

23

20

32

60

14

00

14

06

14

18

14

24

14

32

14

55

16

48

14

62

14

72

14

73

14

74

14

79

19

40

21

76

14

80

14

86

14

92

14

98

15

02

15

03

15

08

15

14

20

60

23

08

15

24

15

26

15

34

15

44

15

52

15

56

15

57

15

58

15

61

20

00

15

86

15

91

15

98

16

12

16

17

16

28

16

34

16

36

23

96

16

53

16

58

19

06

16

64

16

65

16

88

16

94

17

12

17

14

17

18

17

24

17

26

17

29

17

32

17

42

17

48

17

66

28

36

17

84

17

98

18

02

18

04

18

07

18

08

18

22

18

33

26

24

27

10

18

40

18

49

18

62

18

69

20

18

18

70

18

81

25

40

18

84

18

86

18

92

18

88

19

03

19

12

19

17

19

18

24

14

19

42

31

70

25

54

19

60

19

64

19

66

19

72

19

77

19

82

19

89

21

34

20

06

20

13

20

25

20

26

20

31

20

36

20

38

35

48

26

68

20

52

20

54

20

68

24

88

21

92

20

72 20

86

22

30

20

90

20

92

20

98

21

03

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14

21

44

21

50

21

55

21

58

21

68

21

82

21

86

21

88

34

46

21

96

22

41

22

05

22

18

22

29

22

42

22

46

22

48

22

76

22

78

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95

29

08

23

00

36

28

23

38

23

42

23

50

23

54

23

59

23

66

23

67

23

72

24

13

31

66

24

20

24

32

24

62

24

63

24

74

24

82

24

98

25

10

25

18

26

54

25

30

25

46

28

48

28

22

25

52

25

53

25

66

25

78

25

85

28

82

25

90

25

94

26

06

26

12

26

18

26

25

26

26

26

50

26

53

26

55

26

66

34

40

47

44

26

80

26

86

26

92

27

08

27

15

30

70

27

27

30

32

27

28

27

34

27

51

27

52

27

68

28

06

28

34

28

39

28

46

28

58

41

96

31

54

28

60

29

14

29

15

29

18

29

19

29

26

29

42

29

45

29

54

29

61

48

76

29

64

29

66

35

12

30

88

29

68

29

78

29

84

30

58

29

90

29

98

30

02

30

08

30

21

30

22

30

28

30

31

30

45

30

46

30

62

30

73

30

74

30

94

30

98

31

04

31

05

31

06

31

11

31

35

31

83

32

01

32

08

32

18

32

62

32

66

32

73

32

74

33

08

33

10

33

22

33

45

33

46

33

62

33

82

45

56

40

12

33

96

34

09

35

02

34

10

34

12

34

17

34

18

34

28

41

66

34

42

34

48

34

53

34

58

38

48

41

32

34

60

34

64

40

23

34

65

34

81

34

88

35

07

35

21

35

24

35

26

35

31

35

54

35

66

35

67

35

98

36

02

36

09

36

16

36

70

36

82

45

46

37

00

40

66

37

10

37

28

37

30

37

31

37

34

37

78

37

88

37

94

38

18

38

24

38

49

38

68

39

25

39

34

39

58

39

88

39

94

40

06

40

09

40

17

40

33

40

46

40

52

40

53

40

65

41

38

40

70

40

82

41

02

41

74

41

78

42

02

42

03

42

14

42

15

42

22

42

38

42

50

42

62

42

64

42

74

42

82

42

93

42

94

42

98

43

03

43

04

43

18

43

23

43

53

43

58

43

66

43

78

43

82

44

02

44

07

44

19

44

29

44

62

44

66

44

78

44

86

44

97

45

14

45

21

45

26

45

27

45

51

45

68

45

82

45

94

45

98

46

04

46

06

46

11

46

42

46

59

47

26

46

72

46

78

46

87

46

94

47

02

47

11

47

31

47

86

48

08

48

15

48

22

48

32

48

33

48

34

48

35

48

39

48

49

48

50

48

56

48

99

49

18

49

21

49

24

49

42

49

66

49

72

49

78

Figure 12. Prime family 43

Page 14: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

14 RICHARD J. MATHAR

4 7 p

1 2 9 205

2 6 1

459

493

7 2 5

7 7 1

995

1 2 4 5

1 3 7 92 0 7 1 2 1 2 3 24352483

2 7 8 12973

2835

3007

3595

4087 4163

4 5 7 9

4653

Figure 13. Prime family 47

53p

235 4 5 1 667

10851 1 4 5 1589 2453

2 5 4 7

3 1 0 1

3 213

3305

3895

461348294853

Figure 14. Prime family 53

Page 15: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 15

26

5

59

p

20

0

47

6

53

2

25

2

39

4

35

04

70

58

8

51

7

51

8

10

08

22

16

82

4

73

6

61

6

68

26

94

69

7

82

0

94

4

70

4

77

6

75

2

14

84

15

40

75

6

91

68

30

85

0

93

4

93

8

14

14

10

34

98

0

19

54

10

84

11

24

24

92

11

60

14

02

11

90

13

84

12

26

12

20

12

70

13

82

14

36

14

66

15

10

25

48

15

46

16

16

16

42

16

54

19

28

17

02

16

72

16

84

17

30

21

04

18

56

17

36

29

36

25

84

17

76

17

99

18

10

18

50

18

82

19

10

19

46

19

52

19

78

20

30

22

90

20

02

20

14

20

50

20

62

33

92

34

66

20

64

22

54

22

82

23

14

23

18

23

30

23

74

23

78

24

04

24

10

24

34

24

46

30

38

24

94

46

16

40

54

24

96

28

16

33

16

26

62

26

96

27

02

28

66

27

50

27

62

27

86

27

98

28

30

29

38

29

62

30

14

30

44

30

64

30

76

30

86

31

22

32

06

32

26

32

36

32

63

33

38

33

68

33

74

33

98

34

22

34

36

34

55

35

63

35

86

36

32

36

64

36

74

36

94

37

58

37

64

41

36

45

04

37

84

38

12

38

14

38

32

38

38

38

42

38

98

39

02

39

08

39

56

39

98

40

10

40

22

40

76

41

48

41

54

43

1043

84

44

12

44

26

45

02

45

52

45

70

45

74

46

12

46

22

46

88

47

06

47

38

47

42

47

72

49

16

49

90

Figure 15. Prime family 59

Page 16: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

16 RICHARD J. MATHAR

15

0

22

2

13

8

23

4

31

2

52

8

16

8

96

0

20

88

37

62

55

98

45

6

74

4

26

4

11

76

22

44

38

04

51

00

Figure 16. Other branches of prime family 59

2 0 1

7 1 p

1 6 2 423 5 9 1

649 901 1 0 8 1 1 1 8 9

1 2 5 7

1 3 5 11659

1 7 6 1

2055 20792967

2500

2 5 7 5 2 7 9 5

2799 2895

3 2 1 5

3321

343939954 4 8 7 4825

Figure 17. Prime family 71

Page 17: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 17

7 3 p

98 1 7 5

384

636

2 1 6

2 7 3

335

876

1 1 9 6

6 7 1 7 6 7

1 1 0 0

1 5 0 4

804

807 1 1 5 6

993

1 0 0 7

1 4 5 2

2 2 7 2

1 0 6 8

1 5 2 0

1 1 6 9

1 2 4 7 1 2 7 1

1 3 0 4

2 5 1 6

1 4 0 4

2 2 6 4

2 2 0 0

3380

1 5 3 21 5 9 3

3 3 5 2

2 9 4 8

1 7 2 8

2 7 7 6

2 4 4 4

1 7 6 0

1 7 9 3

2 2 6 0

1 8 1 2

2636

1 9 8 4

1 9 5 6

2080

3 2 1 2

1 9 9 6

2 0 2 0

3004

2096

2 1 3 2

2 6 3 2

3 1 2 8

2 1 5 6

2 1 9 4

4306

2 2 0 4

2 5 2 8

2 5 1 2

2306

3 1 1 6

2 7 6 4

2 3 1 6

2 3 3 2

2386

2388

2 6 4 8

2 3 9 2

2 8 7 2

2 4 0 8

2 5 9 6

2 6 0 2

2 6 5 6

2669

2890

3896

3 4 2 4

3304

3399

3 4 5 2

3 4 7 3

3 5 3 2

3 5 4 4

3 5 5 3

3590

3626

3638

3 6 7 6

3686

3805

3 8 5 4

3928

3986

4 0 2 8

4 0 3 4

4030

4 0 9 4

4 1 8 44 2 5 8

4 5 3 8

4 7 6 6

4 7 7 8

4 8 1 0

4 8 7 9

4 8 8 2

4 8 9 2

4 9 0 4

Figure 18. Prime family 73

Page 18: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

18 RICHARD J. MATHAR

7 9 p

365 4 9 7 7 3 7 1 0 3 7 1 1 2 1

2 4 5 5

1296

1 3 7 5

1 4 5 7 1 5 1 7

16291 7 9 5 1 8 5 5

1953 2 1 1 5

2199 2493 29433099 33513883 3955

40594 1 3 7

4 1 9 1

4359

4865

Figure 19. Prime family 79

83p

2 3 7

699

7 8 1 1 3 5 7 1 5 3 7

1603 2899 3059 3 1 2 5

3185

4355

4 837

Figure 20. Prime family 83

89p

1 7 1 4 1 5

5 0 1

7 9 5

1 1 4 1

1 2 0 7 1 7 1 1 1 9 2 7

2 0 4 1 20452869 40254309

4459

Figure 21. Prime family 89

Page 19: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 19

9 7 p

2 4 5 2 7 5 623

7 2 3 813

1 0 7 9

1 1 9 5

1343

1 3 4 5

1 4 9 11 5 8 1

1 6 7 9

1 8 5 7

1943

2 1 5 7

2183 2 2 7 9

2 4 2 7

2 4 4 5

2535

2563

2 5 9 5

2613

2893 3085

3525

3859

4035

4085 4 1 4 9

4369

4 4 6 1

4893

4 6 2 4

Figure 22. Prime family 97

1 0 1 p

2 9 1 979 1 4 1 1

1 9 8 1

2059 2 4 1 9 2 4 9 1

3 6 0 1

3698 4802

Figure 23. Prime family 101

103p

485 1 1 5 7 1 5 7 7 1 8 1 7 2 1 1 7 2 2 0 1

2 395

2 5 0 1 2 5 3 7

3459 4923

Figure 24. Prime family 103

Page 20: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

20 RICHARD J. MATHAR

1 0 7 p

309

3 1 5

505

903

933 1 1 2 5

1 5 1 3

1 5 1 5 1683

1909

1 9 4 7

1 9 7 5

2 0 9 1 2 1 3 9

2 1 4 9

2495

2 7 4 5

2 7 7 3

2 7 8 7

3 1 7 9

3381 3597

3 7 5 1 3887

4235

4 2 5 1

4 5 3 3 4605

4 7 4 3

Figure 25. Prime family 107

109p

325 5 1 5 7 0 7 1 0 6 7 1 6 9 1

21332 2 1 9

2 2 9 1

2 5 4 5

2 6 2 7 2 7 4 7 2867

3075

3189 34293443 3505

38254 0 1 1

4 0 4 3

4 5 8 7

Figure 26. Prime family 109

1 1 3 p

3 2 7 535 1 1 1 1 2 0 4 7

2048

2 4 0 7 2 9 1 1 3 1 2 7

4069

Figure 27. Prime family 113

1 2 7 p

1 2 8 1469 1853 2033 2369 2813 3293 3569 3 7 1 3 3869 3953

Figure 28. Prime family 127

Page 21: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 21

1 3 1 p

189 381

900

1 9 2 1

738

1 2 6 7

2 4 6 1

2 6 1 1

2929 3649 3901 4189

4 7 7 14 8 6 7

Figure 29. Prime family 131

1 3 7 p

2 7 9 655

1 8 9 7

2599 3103 4183 4399

4 503

4529

Figure 30. Prime family 137

139p

9 1 7 1 3 9 7 3 1 6 1 3 3 1 7 3 7 3 7 3 9 7 7

4 5 5 5

4661 4 7 5 7

Figure 31. Prime family 139

1 4 9 p

1 5 0 7 2 2 2 7 4387 4 7 4 7

Figure 32. Prime family 149

Page 22: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

22 RICHARD J. MATHAR

1 5 1 p

1 5 2 9 1 7 8 1 2489 2 9 2 1

3 4 1 1

3885

4 1 8 1 4469

4 5 4 5

4601 4 8 4 1

Figure 33. Prime family 151

1 5 7 p

2 4 2

4 7 8

628

7 5 5 1043

2253

2363 2603

3 1 1 7 3565

3683

3795

4845

4859

Figure 34. Prime family 157

Page 23: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 23

163p

7 8 5 1 6 6 1

1 7 3 7

1 9 3 7

2 7 7 5 3159

3 1 9 7

3 879

4061

4 9 7 1

294

390

282

618

630

486

606

4 7 4

1 2 4 2

1638

906

918

894

1626

1 1 1 0

2730

1 2 5 4 1 6 1 4

5334

Figure 35. Prime family 163

1 6 7 p

489 2533 3973

Figure 36. Prime family 167

1 7 3 p

835

1533

3 4 2 7

4 1 4 5

Figure 37. Prime family 173

Page 24: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

24 RICHARD J. MATHAR

1 7 9 p

865 1 8 3 7

3 4 7 5 4295

4 3 2 1

4949

Figure 38. Prime family 179

1 8 1 p

1 2 1 1

1 7 0 1

2 1 7 1 2 7 7 1 3 6 1 1

3685

4379 4619

Figure 39. Prime family 181

1 9 1 p

385

1 4 7 5 1895

1969

2009

2 9 4 1

4103

3136

3841

4 4 1 3 4 4 9 1

Figure 40. Prime family 191

193p

605

676

688

833

892

908

1 3 4 6

1 6 4 71803

1 9 9 1 2 3 2 7

2 4 8 72995

3285

3287

3915

4 1 3 5 4 1 6 5

4 7 2 7

4 7 5 3

Figure 41. Prime family 193

Page 25: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 25

1 9 7 p

5 7 9 955 3043 3979

3 9 9 7

4843

Figure 42. Prime family 197

199p

965 1 3 3 7

2979

3 0 7 7 3401

Figure 43. Prime family 199

2 1 1 p

338 2189 2 5 6 1 3281 3629

4 9 4 1

Figure 44. Prime family 211

284

220

562

Figure 45. Aliquot cycle 220

223p

2 3 2 1 4 5 7 7

Figure 46. Prime family 223

Page 26: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

26 RICHARD J. MATHAR

2 2 7 p

669

4 6 2 7

Figure 47. Prime family 227

229p

1 1 1 5 3587

Figure 48. Prime family 229

233p

687 1 1 3 5

2049

4 5 8 1

Figure 49. Prime family 233

7 7 4

942

7 6 2

954

1 1 5 2

2163

1 1 6 5

239p

2 4 9 7

Figure 50. Prime family 239

Page 27: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 27

2 4 1 p

399 1 1 2 7 1 6 3 1 2 5 1 9

2 9 0 5

2 9 5 1

3369

3 7 9 1

4881

Figure 51. Prime family 241

Page 28: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

28 RICHARD J. MATHAR

2 5 1 p

2629 3961

Figure 52. Prime family 251

2 5 7 p

549

987 1 1 3 1

1 2 5 5

1605

1 9 7 6

2 2 2 4

1 6 2 4

1 7 5 5

2 1 1 6

2499

2949

2956

3 2 4 2

3688

3723

3 7 8 7

39323946

4063

4226

4232

4 2 9 2

4 4 4 2

4697

4 7 7 4

4730

4803

4910

Figure 53. Prime family 257

263p

1 2 8 5 2 7 6 1

Figure 54. Prime family 263

Page 29: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 29

269p

595 1 3 1 5

2093

2 8 2 7

28884109

4 2 6 7

Figure 55. Prime family 269

2 7 1 p

1 8 4 1 3341

3483 4 1 1 3

4769

Figure 56. Prime family 271

396

696

2 7 6 306

1 1 0 4

1 8 7 2

3 7 7 0

3790

2 7 7 2

5964

2 7 1 6

2864 3050

3430

Figure 57. Sequence of 276 with unknown termination.

Page 30: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

30 RICHARD J. MATHAR

2 7 7 p

1 0 2 5 1 3 5 5 1883

3063

3419 4883

Figure 58. Prime family 277

2 8 1 p

603 831 1 1 8 3

1 2 2 1 1 7 9 7 2 4 8 1

2955

2959

3 651

4 4 7 1

4 7 7 9

Figure 59. Prime family 281

283p

1385 2981 3497

4 1 4 3

4997

Figure 60. Prime family 283

293p

7 1 5 3091

3545

Figure 61. Prime family 293

307p

3809 4913

Figure 62. Prime family 307

Page 31: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 31

3 1 1 p

9 2 1

2043

4981

Figure 63. Prime family 311

313p

1 1 7 5 1 5 3 5

3513 4593

Figure 64. Prime family 313

3 1 7 p

939 1 5 5 5

Figure 65. Prime family 317

331p

4 1 2 1

Figure 66. Prime family 331

337p

1 6 5 5 1859 2 0 5 7

Figure 67. Prime family 337

3 4 7 p

805

Figure 68. Prime family 347

Page 32: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

32 RICHARD J. MATHAR

523p

324

Figure 69. Prime family 523

Page 33: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 33

366

378

180 2 1 0 354

582

594

846

1026

1 3 7 4

6 1 2

834

822

1386

930 1362

2358

2790

2 7 7 8

2 2 6 2

4698

2766

6192

Figure 70. Prime family 601

Page 34: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

34 RICHARD J. MATHAR

240

504

1 2 0

1056

1968

3240

1 2 7 2

7650

Figure 71. Prime family 12161

Page 35: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 35

330

534

318

546

5 1 0

786

498

798

1 1 2 2

1 4 7 0

7 5 0

2634

2646

1 830 1 9 7 4

4 1 9 4

4932

7626

Figure 72. Prime family 321329

Page 36: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

36 RICHARD J. MATHAR

3. Unitary Sigma Illustrations

Mapping n to the sum of the unitary divisors of n [2, 3, 4] as tabulated inA034448 [9] defines another set of merger families, illustrated in Figures 73–77.

Page 37: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 37

3 42 5 6

12

20

8 97 10

18

30

11

17

16

72

19

32

33

21

29

90

31

48

68

35

38

60

37

12

0

56

39

44

43

45

46

47

51

70

14

4

52

55

80

10

2

57

18

0

58

59

21

6

63

17

0

66

67

71

10

0

13

0

76

79

87

89

30

0

92

94

95

99

25

2

10

1

11

0

10

9

15

2

11

11

15

11

8

11

9

40

0

14

2

32

4

14

5

15

1

15

3

15

9

16

61

69

41

0

17

9

52

0

18

4

22

8

18

5

18

7

20

5

21

4

22

12

27

44

2

23

62

45

25

1

26

12

65

44

8

27

3

75

6

31

64

03

40

9

11

20

41

24

75

50

26

64

68

96

97

Figure 73. Unitary sigma family 2

Page 38: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

38 RICHARD J. MATHAR

14

24

13

36

15

50

22

23

78

49

16

8

28

8

12

31

35

13

8

13

7

33

0

14

3

15

41

65

16

7

24

8

18

3

21

3

86

4

21

8

23

52

53

Figure 74. Unitary sigma family 13

Page 39: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 39

26

42

25

96

41

13

2

62

61

69

74

11

4

73

24

0

10

4

12

6

75

77

8281

10

8

14

0

85

86

88

10

31

07

11

3

40

8

11

71

25

13

11

39

15

8

15

7

17

71

88

20

32

07

20

92

39

64

8

30

3

31

8

31

7

73

8

44

0

32

73

35

36

8

36

7

37

4

37

33

87

43

9

53

55

68

58

36

47

12

60

Figure 75. Unitary sigma family 25

Page 40: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

40 RICHARD J. MATHAR

28

40

27

54

84

34

53

16

0

65

64

83

19

8

12

8

12

9

93

12

4

12

7

17

6

20

4

13

3

36

0

13

4

19

0

14

8

22

0

17

2

17

4

17

3

54

0

19

7

27

2

30

6

20

1

20

23

04

34

0

25

9

26

72

68

27

1

28

4

28

3

29

0

28

9

84

0

29

53

19

32

3

35

8

35

9

42

84

45

47

24

77

49

3

Figure 76. Unitary sigma family 27

Page 41: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

ALIQUOT SEQUENCE MERGERS 41

11

2

13

6

91

98

15

0

97

31

2

16

2

24

6

10

61

16

22

2

45

6

14

6

14

9

50

4

28

0

43

2

15

62

08

23

8

17

5

27

0

17

8

22

4

26

4

18

9

20

62

12

21

12

15

72

0

22

3

23

0

22

9

47

6

23

2

24

7

25

5

26

3

26

9

41

4

27

4

27

5

28

63

02

31

13

21

33

43

39

35

4

35

3

10

20

35

53

71

37

5

37

63

91

40

6

40

74

15

41

8

43

1

43

5

45

14

59

46

06

32

47

1

47

84

95

50

3

53

76

23

63

1

63

96

49

66

77

19

Figure 77. Unitary sigma family 91

Page 42: ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 · ILLUSTRATION OF ALIQUOT SEQUENCE MERGERS RICHARD J. MATHAR Abstract. Aliquot sequences are

42 RICHARD J. MATHAR

References

1. Manuel Benito and Juan L. Varona, Advances in aliquot cycles, Math. Comp. 68 (1999),no. 225, 389–393. MR 1489967 (99c:11162)

2. Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math.

Zeitschr. 74 (1960), 66–80. MR 0112861 (22#3707)3. , The number of unitary divisors of an integer, Am. Math. Monthly 67 (1960), no. 9,

879–880. MR 0122790 (23#A124)

4. Graeme L. Cohen, On an integer’s infinitary divisors, Math. Comp. 54 (1990), no. 189, 395–411. MR 0993927 (90e:11011)

5. P. Erdos, On asymptotic properties of aliquot sequences, Math. Comp. 30 (1976), no. 135,641–645. MR 0404115 (53#7919)

6. Richard K. Guy, Unsolved problems in number theory, 3 ed., Springer, Heidelberg, 2004.

MR 2076335 (2005h:11003)7. Richard K. Guy and J. L. Selfridge, What drives an aliquot sequence?, Math. Comp. 29 (1975),

no. 129, 101–107. MR 0384669 (52#5542)

8. David Moews and Paul C. Moews, A search for aliquot cycles and amicable pairs, Math. Comp.61 (1993), no. 204, 935–938. MR 1185249 (94a:11198)

9. Neil J. A. Sloane, The On-Line Encyclopedia Of Integer Sequences, Notices Am. Math. Soc.

50 (2003), no. 8, 912–915, http://oeis.org/. MR 1992789 (2004f:11151)

URL: http://www.mpia.de/~mathar

Hoeschstr. 7, 52372 Kreuzau, Germany