ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 ·...
Transcript of ILLUSTRATION OF ALIQUOT SEQUENCE MERGERSmathar/public/mathar20100305.pdf · 2014-03-21 ·...
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
2 RICHARD J. MATHAR
Figure 1. Prime family 3 [9, A127163]
3p49
16
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ALIQUOT SEQUENCE MERGERS 3
Figure 2. Aliquot cycle 6.
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4 RICHARD J. MATHAR
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Figure 3. Prime family 7 [9, A127164].
ALIQUOT SEQUENCE MERGERS 5
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Figure 4. Prime family 11
6 RICHARD J. MATHAR
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Figure 5. Prime family 13
ALIQUOT SEQUENCE MERGERS 7
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Figure 6. Prime family 17
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Figure 8. Prime family 23
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Figure 9. Prime family 31
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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
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
8
37
0
37
7
46
0
54
8
38
0
38
7
40
6
67
2
13
44
40
8
41
2
41
8
42
84
34
43
6
43
7
44
1
62
0
72
4
44
4
44
64
47
44
8
45
8
80
6 53
8
46
8
47
24
82
48
4
48
8
53
6
49
0
11
36
49
55
06
51
1
51
4
71
6
54
4
51
65
30
54
3
59
0
56
6
55
0
92
8
96
2
56
0
60
4
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
21
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
22
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
32 RICHARD J. MATHAR
523p
324
Figure 69. Prime family 523
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
34 RICHARD J. MATHAR
240
504
1 2 0
1056
1968
3240
1 2 7 2
7650
Figure 71. Prime family 12161
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
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.
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
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
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
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
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
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
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