Representation of n based on its factorization into maximal prime powers

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A representation of n based on its factorization into maximal prime powers with prime powers increasing rightward (by analogy with base b representation)

{0, 1, 10, 100, 1000, 11, 10000, 100000, 1000000, 1001, 10000000, 110, 100000000, 10001, 1010, 1000000000, 10000000000, 1000001, 100000000000, 1100, 10010, 10000001, ...}

which may be represented in base 8 by grouping triples of binary digits into single octal digits

{0, 1, 2, 4, 10, 3, 20, 40, 100, 11, 200, 6, 400, 21, 12, 1000, 2000, 101, 4000, 14, 22, 201, 10000, 42, 20000, 401, 40000, 24, 100000, 13, 200000, ...}

is given in the following table. Note that although the representation is very economical in the set of digits needed, i.e. {0, 1} for the binary version, it is extremely uneconomical in the number of digits required, up to the number of prime powers up to n, which is asymptotic to the number of primes up to n, i.e. $\scriptstyle {\frac {n}{\log n}}\,$, making this representation absolutely impractical! (Cf. A025528)

Now, if one wants to create an ordering of the positive integers by increasing values of representation of n based on its factorization into maximal prime powers, we have the annoying issue of dealing with inadmissible values for the representation, e.g. 101 would represent (rightwards) 8 as 2 times 4, which are not maximal prime powers in the factorization of 8. We do not have a one-to-one and onto correspondence between the positive integers (as products of maximal prime powers) and the nonnegative integers (as representation of n based on its factorization into maximal prime powers.) But there is a solution to this: by considering the exponents of prime powers to be in base 2 representation, which amounts to considering only prime powers with powers of two as exponents.

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

Table of representation of n based on its factorization into maximal prime powers

Factorization of n into maximal prime powers

$n\,$	127	125	121	113	109	107	103	101	97	89	83	81	79	73	71	67	64	61	59	53	49	47	43	41	37	32	31	29	27	25	23	19	17	16	13	11	9	8	7	5	4	3	2	Base 8
1																																											0	$\scriptstyle 0\,$
2																																											1	$\scriptstyle 1\,$
3																																										1	0	$\scriptstyle 2\,$
4																																									1	0	0	$\scriptstyle 4\,$
5																																								1	0	0	0	$\scriptstyle 10\,$
6																																										1	1	$\scriptstyle 3\,$
7																																							1	0	0	0	0	$\scriptstyle 20\,$
8																																						1	0	0	0	0	0	$\scriptstyle 40\,$
9																																					1	0	0	0	0	0	0	$\scriptstyle 100\,$
10																																								1	0	0	1	$\scriptstyle 11\,$
11																																				1	0	0	0	0	0	0	0	$\scriptstyle 200\,$
12																																									1	1	0	$\scriptstyle 6\,$
13																																			1	0	0	0	0	0	0	0	0	$\scriptstyle 400\,$
14																																							1	0	0	0	1	$\scriptstyle 21\,$
15																																								1	0	1	0	$\scriptstyle 12\,$
16																																		1	0	0	0	0	0	0	0	0	0	$\scriptstyle 1000\,$
17																																	1	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 2000\,$
18																																					1	0	0	0	0	0	1	$\scriptstyle 101\,$
19																																1	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 4000\,$
20																																								1	1	0	0	$\scriptstyle 14\,$
21																																							1	0	0	1	0	$\scriptstyle 22\,$
22																																				1	0	0	0	0	0	0	1	$\scriptstyle 201\,$
23																															1	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 10000\,$
24																																						1	0	0	0	1	0	$\scriptstyle 42\,$
25																														1	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 20000\,$
26																																			1	0	0	0	0	0	0	0	1	$\scriptstyle 401\,$
27																													1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 40000\,$
28																																							1	0	1	0	0	$\scriptstyle 24\,$
29																												1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 100000\,$
30																																								1	0	1	1	$\scriptstyle 13\,$
31																											1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 200000\,$
32																										1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 400000\,$
33																																				1	0	0	0	0	0	1	0	$\scriptstyle 202\,$
34																																	1	0	0	0	0	0	0	0	0	0	1	$\scriptstyle 2001\,$
35																																							1	1	0	0	0	$\scriptstyle 30\,$
36																																					1	0	0	0	1	0	0	$\scriptstyle 104\,$
37																									1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 1000000\,$
38																																1	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle 4001\,$
39																																			1	0	0	0	0	0	0	1	0	$\scriptstyle 402\,$
40																																						1	0	1	0	0	0	$\scriptstyle 50\,$
41																								1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 2000000\,$
42																																							1	0	0	1	1	$\scriptstyle 23\,$
43																							1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 4000000\,$
44																																				1	0	0	0	0	1	0	0	$\scriptstyle 204\,$
45																																					1	0	0	1	0	0	0	$\scriptstyle 110\,$
46																															1	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle 10001\,$
47																						1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 10000000\,$
48																																		1	0	0	0	0	0	0	0	1	0	$\scriptstyle 1002\,$
49																					1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 20000000\,$
50																														1	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle 20001\,$
51																																	1	0	0	0	0	0	0	0	0	1	0	$\scriptstyle 2002\,$
52																																			1	0	0	0	0	0	1	0	0	$\scriptstyle 404\,$
53																				1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 40000000\,$
54																													1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle 40001\,$
55																																				1	0	0	0	1	0	0	0	$\scriptstyle 210\,$
56																																						1	1	0	0	0	0	$\scriptstyle 60\,$
57																																1	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle 4002\,$
58																												1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle 100001\,$
59																			1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 100000000\,$
60																																								1	1	1	0	$\scriptstyle 16\,$
61																		1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 200000000\,$
62																											1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle 200001\,$
63																																					1	0	1	0	0	0	0	$\scriptstyle 120\,$
64																	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle 400000000\,$
65																																			1	0	0	0	0	1	0	0	0	$\scriptstyle \,$
66																																				1	0	0	0	0	0	1	1	$\scriptstyle \,$
67																1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
68																																	1	0	0	0	0	0	0	0	1	0	0	$\scriptstyle \,$
69																															1	0	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle \,$
70																																							1	1	0	0	1	$\scriptstyle \,$
71															1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
72																																					1	1	0	0	0	0	0	$\scriptstyle \,$
73														1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
74																									1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
75																														1	0	0	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle \,$
76																																1	0	0	0	0	0	0	0	0	1	0	0	$\scriptstyle \,$
77																																				1	0	0	1	0	0	0	0	$\scriptstyle \,$
78																																			1	0	0	0	0	0	0	1	1	$\scriptstyle \,$
79													1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
80																																		1	0	0	0	0	0	1	0	0	0	$\scriptstyle \,$
81												1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
82																								1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
83											1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
84																																							1	0	1	1	0	$\scriptstyle \,$
85																																	1	0	0	0	0	0	0	1	0	0	0	$\scriptstyle \,$
86																							1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
87																												1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle \,$
88																																				1	0	1	0	0	0	0	0	$\scriptstyle \,$
89										1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
90																																					1	0	0	1	0	0	1	$\scriptstyle \,$
91																																			1	0	0	0	1	0	0	0	0	$\scriptstyle \,$
92																															1	0	0	0	0	0	0	0	0	0	1	0	0	$\scriptstyle \,$
93																											1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle \,$
94																						1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
95																																1	0	0	0	0	0	0	0	1	0	0	0	$\scriptstyle \,$
96																										1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle \,$
97									1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
98																					1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
99																																				1	1	0	0	0	0	0	0	$\scriptstyle \,$
100																														1	0	0	0	0	0	0	0	0	0	0	1	0	0	$\scriptstyle \,$
101								1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
102																																	1	0	0	0	0	0	0	0	0	1	1	$\scriptstyle \,$
103							1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
104																																			1	0	0	1	0	0	0	0	0	$\scriptstyle \,$
105																																							1	1	0	1	0	$\scriptstyle \,$
106																				1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
107						1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
108																													1	0	0	0	0	0	0	0	0	0	0	0	1	0	0	$\scriptstyle \,$
109					1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
110																																				1	0	0	0	1	0	0	1	$\scriptstyle \,$
111																									1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle \,$
112																																		1	0	0	0	0	1	0	0	0	0	$\scriptstyle \,$
113				1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
114																																1	0	0	0	0	0	0	0	0	0	1	1	$\scriptstyle \,$
115																															1	0	0	0	0	0	0	0	0	1	0	0	0	$\scriptstyle \,$
116																												1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	$\scriptstyle \,$
117																																			1	0	1	0	0	0	0	0	0	$\scriptstyle \,$
118																			1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
119																																	1	0	0	0	0	0	1	0	0	0	0	$\scriptstyle \,$
120																																						1	0	1	0	1	0	$\scriptstyle \,$
121			1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
122																		1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	$\scriptstyle \,$
123																								1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	$\scriptstyle \,$
124																											1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	$\scriptstyle \,$
125		1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$
126																																					1	0	1	0	0	0	1	$\scriptstyle \,$
127	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	$\scriptstyle \,$

See also

• Prime factorization
• Prime powers
• Representation of n based on its factorization into prime powers with powers of two as exponents