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

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

${\displaystyle p^{a}=p^{{\big \{}\sum _{i=0}^{1+\lfloor \log _{2}(a)\rfloor }a_{i}\cdot 2^{i}{\big \}}},\quad a_{i}\in \{0,1\}\,}$

${\displaystyle =\prod _{i=0}^{1+\lfloor \log _{2}(a)\rfloor }p^{(a_{i}\cdot 2^{i})}\,}$

${\displaystyle =\prod _{i=0}^{1+\lfloor \log _{2}(a)\rfloor }(p^{2^{i}})^{a_{i}}\,}$

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. ${\displaystyle \scriptstyle {\frac {n}{\log n}}\,}$, 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

${\displaystyle n\,}$	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	${\displaystyle \scriptstyle 0\,}$
2																																						1	${\displaystyle \scriptstyle 1\,}$
3																																					1	0	${\displaystyle \scriptstyle 2\,}$
4																																				1	0	0	${\displaystyle \scriptstyle 4\,}$
5																																			1	0	0	0	${\displaystyle \scriptstyle 10\,}$
6																																					1	1	${\displaystyle \scriptstyle 3\,}$
7																																		1	0	0	0	0	${\displaystyle \scriptstyle 20\,}$
8																																				1	0	1	${\displaystyle \scriptstyle 5\,}$
9																																	1	0	0	0	0	0	${\displaystyle \scriptstyle 40\,}$
10																																			1	0	0	1	${\displaystyle \scriptstyle 11\,}$
11																																1	0	0	0	0	0	0	${\displaystyle \scriptstyle 100\,}$
12																																				1	1	0	${\displaystyle \scriptstyle 6\,}$
13																															1	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 200\,}$
14																																		1	0	0	0	1	${\displaystyle \scriptstyle 21\,}$
15																																			1	0	1	0	${\displaystyle \scriptstyle 12\,}$
16																														1	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 400\,}$
17																													1	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 1000\,}$
18																																	1	0	0	0	0	1	${\displaystyle \scriptstyle 41\,}$
19																												1	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 2000\,}$
20																																			1	1	0	0	${\displaystyle \scriptstyle 14\,}$
21																																		1	0	0	1	0	${\displaystyle \scriptstyle 22\,}$
22																																1	0	0	0	0	0	1	${\displaystyle \scriptstyle 101\,}$
23																											1	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 4000\,}$
24																																				1	1	1	${\displaystyle \scriptstyle 7\,}$
25																										1	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 10000\,}$
26																															1	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 201\,}$
27																																	1	0	0	0	1	0	${\displaystyle \scriptstyle 42\,}$
28																																		1	0	1	0	0	${\displaystyle \scriptstyle 24\,}$
29																									1	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 20000\,}$
30																																			1	0	1	1	${\displaystyle \scriptstyle 13\,}$
31																								1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 40000\,}$
32																														1	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 401\,}$
33																																1	0	0	0	0	1	0	${\displaystyle \scriptstyle 102\,}$
34																													1	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 1001\,}$
35																																		1	1	0	0	0	${\displaystyle \scriptstyle 30\,}$
36																																	1	0	0	1	0	0	${\displaystyle \scriptstyle 44\,}$
37																							1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 100000\,}$
38																												1	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 2001\,}$
39																															1	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 202\,}$
40																																			1	1	0	1	${\displaystyle \scriptstyle 15\,}$
41																						1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 200000\,}$
42																																		1	0	0	1	1	${\displaystyle \scriptstyle 23\,}$
43																					1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 400000\,}$
44																																1	0	0	0	1	0	0	${\displaystyle \scriptstyle 104\,}$
45																																	1	0	1	0	0	0	${\displaystyle \scriptstyle 50\,}$
46																											1	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 4001\,}$
47																				1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 1000000\,}$
48																														1	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 402\,}$
49																			1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 2000000\,}$
50																										1	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 10001\,}$
51																													1	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 1002\,}$
52																															1	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 204\,}$
53																		1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 4000000\,}$
54																																	1	0	0	0	1	1	${\displaystyle \scriptstyle 43\,}$
55																																1	0	0	1	0	0	0	${\displaystyle \scriptstyle 110\,}$
56																																		1	0	1	0	1	${\displaystyle \scriptstyle 25\,}$
57																												1	0	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 2002\,}$
58																									1	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 20001\,}$
59																	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 10000000\,}$
60																																			1	1	1	0	${\displaystyle \scriptstyle 16\,}$
61																1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	${\displaystyle \scriptstyle 20000000\,}$
62																								1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 40001\,}$
63																																	1	1	0	0	0	0	${\displaystyle \scriptstyle 60\,}$
64																														1	0	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 404\,}$
65																															1	0	0	0	1	0	0	0	${\displaystyle \scriptstyle 210\,}$
66																																1	0	0	0	0	1	1	${\displaystyle \scriptstyle 103\,}$
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	${\displaystyle \scriptstyle 40000000\,}$
68																													1	0	0	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 1004\,}$
69																											1	0	0	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 4002\,}$
70																																		1	1	0	0	1	${\displaystyle \scriptstyle 31\,}$
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	${\displaystyle \scriptstyle 100000000\,}$
72																																	1	0	0	1	0	1	${\displaystyle \scriptstyle 45\,}$
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	${\displaystyle \scriptstyle 200000000\,}$
74																							1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 100001\,}$
75																										1	0	0	0	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 10002\,}$
76																												1	0	0	0	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 2004\,}$
77																																1	0	1	0	0	0	0	${\displaystyle \scriptstyle 120\,}$
78																															1	0	0	0	0	0	1	1	${\displaystyle \scriptstyle 203\,}$
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	${\displaystyle \scriptstyle 400000000\,}$
80																														1	0	0	0	0	1	0	0	0	${\displaystyle \scriptstyle 410\,}$
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	${\displaystyle \scriptstyle 1000000000\,}$
82																						1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 200001\,}$
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	${\displaystyle \scriptstyle 2000000000\,}$
84																																		1	0	1	1	0	${\displaystyle \scriptstyle 26\,}$
85																													1	0	0	0	0	0	1	0	0	0	${\displaystyle \scriptstyle 1010\,}$
86																					1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 400001\,}$
87																									1	0	0	0	0	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 20002\,}$
88																																1	0	0	0	1	0	1	${\displaystyle \scriptstyle 105\,}$
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	${\displaystyle \scriptstyle 4000000000\,}$
90																																	1	0	1	0	0	1	${\displaystyle \scriptstyle 51\,}$
91																															1	0	0	1	0	0	0	0	${\displaystyle \scriptstyle 220\,}$
92																											1	0	0	0	0	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 4004\,}$
93																								1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 40002\,}$
94																				1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 1000001\,}$
95																												1	0	0	0	0	0	0	1	0	0	0	${\displaystyle \scriptstyle 2010\,}$
96																														1	0	0	0	0	0	0	1	1	${\displaystyle \scriptstyle 403\,}$
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	${\displaystyle \scriptstyle 10000000000\,}$
98																			1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 2000001\,}$
99																																1	1	0	0	0	0	0	${\displaystyle \scriptstyle 140\,}$
100																										1	0	0	0	0	0	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 10004\,}$
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	${\displaystyle \scriptstyle 20000000000\,}$
102																													1	0	0	0	0	0	0	0	1	1	${\displaystyle \scriptstyle 1003\,}$
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	${\displaystyle \scriptstyle 40000000000\,}$
104																															1	0	0	0	0	1	0	1	${\displaystyle \scriptstyle 205\,}$
105																																		1	1	0	1	0	${\displaystyle \scriptstyle 32\,}$
106																		1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 4000001\,}$
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	${\displaystyle \scriptstyle 100000000000\,}$
108																																	1	0	0	1	1	0	${\displaystyle \scriptstyle 46\,}$
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	${\displaystyle \scriptstyle 200000000000\,}$
110																																1	0	0	1	0	0	1	${\displaystyle \scriptstyle 111\,}$
111																							1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 100002\,}$
112																														1	0	0	0	1	0	0	0	0	${\displaystyle \scriptstyle 420\,}$
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	${\displaystyle \scriptstyle 400000000000\,}$
114																												1	0	0	0	0	0	0	0	0	1	1	${\displaystyle \scriptstyle 2003\,}$
115																											1	0	0	0	0	0	0	0	1	0	0	0	${\displaystyle \scriptstyle 4010\,}$
116																									1	0	0	0	0	0	0	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 20004\,}$
117																															1	0	1	0	0	0	0	0	${\displaystyle \scriptstyle 240\,}$
118																	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 10000001\,}$
119																													1	0	0	0	0	1	0	0	0	0	${\displaystyle \scriptstyle 1020\,}$
120																																			1	1	1	1	${\displaystyle \scriptstyle 17\,}$
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	${\displaystyle \scriptstyle 1000000000000\,}$
122																1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	${\displaystyle \scriptstyle 20000001\,}$
123																						1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	${\displaystyle \scriptstyle 200002\,}$
124																								1	0	0	0	0	0	0	0	0	0	0	0	1	0	0	${\displaystyle \scriptstyle 40004\,}$
125																										1	0	0	0	0	0	0	0	0	1	0	0	0	${\displaystyle \scriptstyle 10010\,}$
126																																	1	1	0	0	0	1	${\displaystyle \scriptstyle 61\,}$
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	${\displaystyle \scriptstyle 2000000000000\,}$

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

• Prime factorization
• Prime powers
• 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 of n based on its factorization into maximal prime powers

References

• S. Finch, "Unitarism and infinitarism" (2004), available at http://www.people.fas.harvard.edu/~sfinch/csolve/try.pdf