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Orderings of integers

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Orderings of integers are ways to put the integers into an order.

Orderings of positive integers

Ordering of positive integers by increasing representation based on their factorization into prime powers with powers of two as exponents

The ordering of positive integers by increasing representation based on their factorization into prime powers with powers of two as exponents (ordering of positive integers by increasing "Fermi-Dirac representation") gives the sequence of integers (a permutation of the positive integers):

Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms. (Cf. A052330)

{1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 63, 126, 189, 378, 252, 504, 756, 1512, 315, 630, 945, 1890, 1260, 2520, 3780, 7560, ...}

Cf. Representation of n based on its factorization into prime powers with powers of two as exponents.

Ordering of positive integers by increasing representation based on their factorization into prime powers with powers of two as exponents
Representation 127 121 113 109 107 103 101 97 89 83 81 79 73 71 67 61 59 53 49 47 43 41 37 31 29 25 23 19 17 16 13 11 9 7 5 4 3 2
0 0
1 1
2 1 0
3 1 1
4 1 0 0
5 1 0 1
6 1 1 0
7 1 1 1
8 1 0 0 0
9 1 0 0 1
10 1 0 1 0
11 1 0 1 1
12 1 1 0 0
13 1 1 0 1
14 1 1 1 0
15 1 1 1 1
16 1 0 0 0 0
17 1 0 0 0 1
18 1 0 0 1 0
19 1 0 0 1 1
20 1 0 1 0 0
21 1 0 1 0 1
22 1 0 1 1 0
23 1 0 1 1 1
24 1 1 0 0 0
25 1 1 0 0 1
26 1 1 0 1 0
27 1 1 0 1 1
28 1 1 1 0 0
29 1 1 1 0 1
30 1 1 1 1 0
31 1 1 1 1 1
32 1 0 0 0 0 0
33 1 0 0 0 0 1
34 1 0 0 0 1 0
35 1 0 0 0 1 1
36 1 0 0 1 0 0
37 1 0 0 1 0 1
38 1 0 0 1 1 0
39 1 0 0 1 1 1
40 1 0 1 0 0 0
41 1 0 1 0 0 1
42 1 0 1 0 1 0
43 1 0 1 0 1 1
44 1 0 1 1 0 0
45 1 0 1 1 0 1
46 1 0 1 1 1 0
47 1 0 1 1 1 1
48 1 1 0 0 0 0
49 1 1 0 0 0 1
50 1 1 0 0 1 0
51 1 1 0 0 1 1
52 1 1 0 1 0 0
53 1 1 0 1 0 1
54 1 1 0 1 1 0
55 1 1 0 1 1 1
56 1 1 1 0 0 0
57 1 1 1 0 0 1
58 1 1 1 0 1 0
59 1 1 1 0 1 1
60 1 1 1 1 0 0
61 1 1 1 1 0 1
62 1 1 1 1 1 0
63 1 1 1 1 1 1
64 1 0 0 0 0 0 0
65 1 0 0 0 0 0 1
66 1 0 0 0 0 1 0
67 1 0 0 0 0 1 1
68 1 0 0 0 1 0 0
69 1 0 0 0 1 0 1
70 1 0 0 0 1 1 0
71 1 0 0 0 1 1 1
72 1 0 0 1 0 0 0
73 1 0 0 1 0 0 1
74 1 0 0 1 0 1 0
75 1 0 0 1 0 1 1
76 1 0 0 1 1 0 0
77 1 0 0 1 1 0 1
78 1 0 0 1 1 1 0
79 1 0 0 1 1 1 1
80 1 0 1 0 0 0 0
81 1 0 1 0 0 0 1
82 1 0 1 0 0 1 0
83 1 0 1 0 0 1 1
84 1 0 1 0 1 0 0
85 1 0 1 0 1 0 1
86 1 0 1 0 1 1 0
87 1 0 1 0 1 1 1
88 1 0 1 1 0 0 0
89 1 0 1 1 0 0 1
90 1 0 1 1 0 1 0
91 1 0 1 1 0 1 1
92 1 0 1 1 1 0 0
93 1 0 1 1 1 0 1
94 1 0 1 1 1 1 0
95 1 0 1 1 1 1 1
96 1 1 0 0 0 0 0
97 1 1 0 0 0 0 1
98 1 1 0 0 0 1 0
99 1 1 0 0 0 1 1
100 1 1 0 0 1 0 0
101 1 1 0 0 1 0 1
102 1 1 0 0 1 1 0
103 1 1 0 0 1 1 1
104 1 1 0 1 0 0 0
105 1 1 0 1 0 0 1
106 1 1 0 1 0 1 0
107 1 1 0 1 0 1 1
108 1 1 0 1 1 0 0
109 1 1 0 1 1 0 1
110 1 1 0 1 1 1 0
111 1 1 0 1 1 1 1
112 1 1 1 0 0 0 0
113 1 1 1 0 0 0 1
114 1 1 1 0 0 1 0
115 1 1 1 0 0 1 1
116 1 1 1 0 1 0 0
117 1 1 1 0 1 0 1
118 1 1 1 0 1 1 0
119 1 1 1 0 1 1 1
120 1 1 1 1 0 0 0
121 1 1 1 1 0 0 1
122 1 1 1 1 0 1 0
123 1 1 1 1 0 1 1
124 1 1 1 1 1 0 0
125 1 1 1 1 1 0 1
126 1 1 1 1 1 1 0
127 1 1 1 1 1 1 1


See also

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