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

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Prime powers with powers of two as exponents

The prime powers with powers of two as exponents might be viewed as "Fermi-Dirac primes" since

where each prime powers with powers of two as exponents thus appears at most once in the "Fermi-Dirac factorization" of n.

"Fermi-Dirac factorization" of n

Related concepts are (necessarily "Fermi-Dirac squarefree") (Cf. A176699)

"Fermi-Dirac composites", "Fermi-Dirac biprimes", "Fermi-Dirac triprimes", ...

"Fermi-Dirac representation" of n

The "Fermi-Dirac factorization" of n suggests a "Fermi-Dirac representation" of n, with prime powers with powers of two as exponents increasing rightward (by analogy with base b representation)

{0, 1, 10, 100, 1000, 11, 10000, 101, 100000, 1001, 1000000, 110, 10000000, 10001, 1010, 100000000, 1000000000, 100001, 10000000000, 1100, 10010, 1000001, 100000000000, 111, ...}

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

{0, 1, 2, 4, 10, 3, 20, 5, 40, 11, 100, 6, 200, 21, 12, 400, 1000, 41, 2000, 14, 22, 101, 4000, 7, 10000, 201, 42, 24, 20000, 13, 40000, 401, 102, 1001, 30, 44, 100000, 2001, ...}

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 with powers of two as exponents up to n, which is asymptotic to the number of primes up to n, i.e. , making this representation absolutely impractical! (Cf. A??????)

Two numbers m and n might be said to be "Fermi-Dirac orthogonal" or "Fermi-Dirac coprime" if they don't share any "Fermi-Dirac prime", i.e.

FDR(m) & FDR(n) = 0, where FDR(n) stands for the "Fermi-Dirac representation" of n and & is the bitwise AND operation.

The product of two "Fermi-Dirac orthogonal" numbers is

FDR(mn) = FDR(m) | FDR(n), when FDR(m) & FDR(n) = 0

where | is the bitwise OR operation.

If one wants to create an ordering of positive integers by increasing values of representation of n based on its factorization into prime powers with powers of two as exponents, we now do have a one-to-one and onto correspondence between the positive integers (as products of prime powers with powers of two as exponents) and the nonnegative integers (as 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 prime powers with powers of two as exponents

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


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

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

Sequences

  • A050376: Numbers of the form p^(2^k) where p is prime and k >= 0.
{2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, ...}
  • A052331: Inverse of A052330. This sequence is a binary representation of this factorization, with a(p^(2^k)) = 2^i, where i is the index of p^(2^k) in A050376.
{0, 1, 2, 4, 8, 3, 16, 5, 32, 9, 64, 6, 128, 17, 10, 256, 512, 33, 1024, 12, 18, 65, 2048, 7, 4096, ...}

See also

References