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	$n\,$
0																																						0	$\scriptstyle 1\,$
1																																						1	$\scriptstyle 2\,$
2																																					1	0	$\scriptstyle 3\,$
3																																					1	1	$\scriptstyle 6\,$
4																																				1	0	0	$\scriptstyle 4\,$
5																																				1	0	1	$\scriptstyle 8\,$
6																																				1	1	0	$\scriptstyle 12\,$
7																																				1	1	1	$\scriptstyle 24\,$
8																																			1	0	0	0	$\scriptstyle 5\,$
9																																			1	0	0	1	$\scriptstyle 10\,$
10																																			1	0	1	0	$\scriptstyle 15\,$
11																																			1	0	1	1	$\scriptstyle 30\,$
12																																			1	1	0	0	$\scriptstyle 20\,$
13																																			1	1	0	1	$\scriptstyle 40\,$
14																																			1	1	1	0	$\scriptstyle 60\,$
15																																			1	1	1	1	$\scriptstyle 120\,$
16																																		1	0	0	0	0	$\scriptstyle 7\,$
17																																		1	0	0	0	1	$\scriptstyle 14\,$
18																																		1	0	0	1	0	$\scriptstyle 21\,$
19																																		1	0	0	1	1	$\scriptstyle 42\,$
20																																		1	0	1	0	0	$\scriptstyle 28\,$
21																																		1	0	1	0	1	$\scriptstyle 56\,$
22																																		1	0	1	1	0	$\scriptstyle 84\,$
23																																		1	0	1	1	1	$\scriptstyle 168\,$
24																																		1	1	0	0	0	$\scriptstyle 35\,$
25																																		1	1	0	0	1	$\scriptstyle 70\,$
26																																		1	1	0	1	0	$\scriptstyle 105\,$
27																																		1	1	0	1	1	$\scriptstyle 210\,$
28																																		1	1	1	0	0	$\scriptstyle 140\,$
29																																		1	1	1	0	1	$\scriptstyle 280\,$
30																																		1	1	1	1	0	$\scriptstyle 420\,$
31																																		1	1	1	1	1	$\scriptstyle 840\,$
32																																	1	0	0	0	0	0	$\scriptstyle 9\,$
33																																	1	0	0	0	0	1	$\scriptstyle 18\,$
34																																	1	0	0	0	1	0	$\scriptstyle 27\,$
35																																	1	0	0	0	1	1	$\scriptstyle 54\,$
36																																	1	0	0	1	0	0	$\scriptstyle 36\,$
37																																	1	0	0	1	0	1	$\scriptstyle 72\,$
38																																	1	0	0	1	1	0	$\scriptstyle 108\,$
39																																	1	0	0	1	1	1	$\scriptstyle 216\,$
40																																	1	0	1	0	0	0	$\scriptstyle 45\,$
41																																	1	0	1	0	0	1	$\scriptstyle 90\,$
42																																	1	0	1	0	1	0	$\scriptstyle 135\,$
43																																	1	0	1	0	1	1	$\scriptstyle 270\,$
44																																	1	0	1	1	0	0	$\scriptstyle 180\,$
45																																	1	0	1	1	0	1	$\scriptstyle 360\,$
46																																	1	0	1	1	1	0	$\scriptstyle 540\,$
47																																	1	0	1	1	1	1	$\scriptstyle 1080\,$
48																																	1	1	0	0	0	0	$\scriptstyle 63\,$
49																																	1	1	0	0	0	1	$\scriptstyle 126\,$
50																																	1	1	0	0	1	0	$\scriptstyle 189\,$
51																																	1	1	0	0	1	1	$\scriptstyle 378\,$
52																																	1	1	0	1	0	0	$\scriptstyle 252\,$
53																																	1	1	0	1	0	1	$\scriptstyle 504\,$
54																																	1	1	0	1	1	0	$\scriptstyle 756\,$
55																																	1	1	0	1	1	1	$\scriptstyle 1512\,$
56																																	1	1	1	0	0	0	$\scriptstyle 315\,$
57																																	1	1	1	0	0	1	$\scriptstyle 630\,$
58																																	1	1	1	0	1	0	$\scriptstyle 945\,$
59																																	1	1	1	0	1	1	$\scriptstyle 1890\,$
60																																	1	1	1	1	0	0	$\scriptstyle 1260\,$
61																																	1	1	1	1	0	1	$\scriptstyle 2520\,$
62																																	1	1	1	1	1	0	$\scriptstyle 3780\,$
63																																	1	1	1	1	1	1	$\scriptstyle 7560\,$
64																																1	0	0	0	0	0	0	$\scriptstyle 11\,$
65																																1	0	0	0	0	0	1	$\scriptstyle 22\,$
66																																1	0	0	0	0	1	0	$\scriptstyle 33\,$
67																																1	0	0	0	0	1	1	$\scriptstyle 66\,$
68																																1	0	0	0	1	0	0	$\scriptstyle 44\,$
69																																1	0	0	0	1	0	1	$\scriptstyle 88\,$
70																																1	0	0	0	1	1	0	$\scriptstyle 132\,$
71																																1	0	0	0	1	1	1	$\scriptstyle 264\,$
72																																1	0	0	1	0	0	0	$\scriptstyle 55\,$
73																																1	0	0	1	0	0	1	$\scriptstyle 110\,$
74																																1	0	0	1	0	1	0	$\scriptstyle 165\,$
75																																1	0	0	1	0	1	1	$\scriptstyle 330\,$
76																																1	0	0	1	1	0	0	$\scriptstyle 220\,$
77																																1	0	0	1	1	0	1	$\scriptstyle 440\,$
78																																1	0	0	1	1	1	0	$\scriptstyle 660\,$
79																																1	0	0	1	1	1	1	$\scriptstyle 1320\,$
80																																1	0	1	0	0	0	0	$\scriptstyle 77\,$
81																																1	0	1	0	0	0	1	$\scriptstyle 154\,$
82																																1	0	1	0	0	1	0	$\scriptstyle 231\,$
83																																1	0	1	0	0	1	1	$\scriptstyle 462\,$
84																																1	0	1	0	1	0	0	$\scriptstyle 308\,$
85																																1	0	1	0	1	0	1	$\scriptstyle 616\,$
86																																1	0	1	0	1	1	0	$\scriptstyle 924\,$
87																																1	0	1	0	1	1	1	$\scriptstyle 1848\,$
88																																1	0	1	1	0	0	0	$\scriptstyle 385\,$
89																																1	0	1	1	0	0	1	$\scriptstyle 770\,$
90																																1	0	1	1	0	1	0	$\scriptstyle 1155\,$
91																																1	0	1	1	0	1	1	$\scriptstyle 2310\,$
92																																1	0	1	1	1	0	0	$\scriptstyle 1540\,$
93																																1	0	1	1	1	0	1	$\scriptstyle 3080\,$
94																																1	0	1	1	1	1	0	$\scriptstyle 4620\,$
95																																1	0	1	1	1	1	1	$\scriptstyle 9240\,$
96																																1	1	0	0	0	0	0	$\scriptstyle 99\,$
97																																1	1	0	0	0	0	1	$\scriptstyle 198\,$
98																																1	1	0	0	0	1	0	$\scriptstyle 297\,$
99																																1	1	0	0	0	1	1	$\scriptstyle 594\,$
100																																1	1	0	0	1	0	0	$\scriptstyle 396\,$
101																																1	1	0	0	1	0	1	$\scriptstyle 792\,$
102																																1	1	0	0	1	1	0	$\scriptstyle 1188\,$
103																																1	1	0	0	1	1	1	$\scriptstyle 2376\,$
104																																1	1	0	1	0	0	0	$\scriptstyle 495\,$
105																																1	1	0	1	0	0	1	$\scriptstyle 990\,$
106																																1	1	0	1	0	1	0	$\scriptstyle 1485\,$
107																																1	1	0	1	0	1	1	$\scriptstyle 2970\,$
108																																1	1	0	1	1	0	0	$\scriptstyle 1980\,$
109																																1	1	0	1	1	0	1	$\scriptstyle 3960\,$
110																																1	1	0	1	1	1	0	$\scriptstyle 5940\,$
111																																1	1	0	1	1	1	1	$\scriptstyle 11880\,$
112																																1	1	1	0	0	0	0	$\scriptstyle 693\,$
113																																1	1	1	0	0	0	1	$\scriptstyle 1386\,$
114																																1	1	1	0	0	1	0	$\scriptstyle 2079\,$
115																																1	1	1	0	0	1	1	$\scriptstyle 4158\,$
116																																1	1	1	0	1	0	0	$\scriptstyle 2772\,$
117																																1	1	1	0	1	0	1	$\scriptstyle 5544\,$
118																																1	1	1	0	1	1	0	$\scriptstyle 8316\,$
119																																1	1	1	0	1	1	1	$\scriptstyle 16632\,$
120																																1	1	1	1	0	0	0	$\scriptstyle 3465\,$
121																																1	1	1	1	0	0	1	$\scriptstyle 6930\,$
122																																1	1	1	1	0	1	0	$\scriptstyle 10395\,$
123																																1	1	1	1	0	1	1	$\scriptstyle 20790\,$
124																																1	1	1	1	1	0	0	$\scriptstyle 13860\,$
125																																1	1	1	1	1	0	1	$\scriptstyle 27720\,$
126																																1	1	1	1	1	1	0	$\scriptstyle 41580\,$
127																																1	1	1	1	1	1	1	$\scriptstyle 83160\,$

See also

• "Fermi-Dirac representation" of n

• Orderings of algebraic numbers and permutation of the algebraic numbers
• Orderings of rational numbers and permutation of the rational numbers
• Orderings of integers and permutation of the integers
• Orderings of positive integers and permutation of the positive integers