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Permutations with three variables:


Permutation table C for x=3 binary variables has 2^x=2^3=8 permutations, ie 000,001,010,011,100,101,110,111.

Table C: binary table for three variables
n
 variable1
 variable2
 variable3
1
 0
 0
 0
2
 0
 0
 1
3
 0
 1
 0
4
 0
 1
 1
5
 1
 0
 0
6
 1
 0
 1
7
 1
 1
 0
8
 1
 1
 1

Proportions:

For the range 1 to 2^x for x=4, for n:
000 occurs 1/(2^x)=1/8 of the time
001 occurs 1/8 of the time
010 occurs 1/8 of the time
011 occurs 1/8 of the time
100 occurs 1/8 of the time
101 occurs 1/8 of the time
110 occurs 1/8 of the time
111 occurs 1/8 of the time

These proportions sum to 1, corresponding to all natural numbers.

1/8+1/8+1/8+1/8+1/8+1/8+1/8+1/8=1. 

Sequences:

The n values where these permutations occur give the four sequences: (only first 4 values shown)

000: n: {1, 9, 17, 25}
001: n: {2, 10, 18, 26}
010: n: {3, 11, 19, 27}
011: n: {4, 12, 20, 28}
100: n: {5, 13, 21, 29}
101: n: {6, 14, 22, 30}
110: n: {7, 15, 23, 31}
111: n: {8, 16, 24, 32}

The difference sequences of the above:
000: Differences[n]: {8,8,8}
001: Differences[n]: {8,8,8}
010: Differences[n]: {8,8,8}
011: Differences[n]: {8,8,8}
100: Differences[n]: {8,8,8}
101: Differences[n]: {8,8,8}
110: Differences[n]: {8,8,8}
111: Differences[n]: {8,8,8}


Permutation table D for x=3 prime divisors has maximum 2^x=2^3=8 distinct permutations, ie
for three prime divisors (2, 3, 5) the divisibility pattern 000,001,010,011,100,101,110,111 gives the 8 distinct permutations in order of first appearance in Table D: 000,100,010,001,110,101,011,111.
Ie. in Table D, for n=3, 3 is divisible by 3 and not 2 and not 5, so the value of the three columns are 0,1,0 for row n=3.

Table D: Prime Factor table for 3 prime factors
n
 prime2
 prime3
 prime5
1
 0
 0
 0
2
 1
 0
 0
3
 0
 1
 0
4
 1
 0
 0
5
 0
 0
 1
6
 1
 1
 0
7
 0
 0
 0
8
 1
 0
 0
9
 0
 1
 0
10
 1
 0
 1
11
 0
 0
 0
12
 1
 1
 0
13
 0
 0
 0
14
 1
 0
 0
15
 0
 1
 1
16
 1
 0
 0
17
 0
 0
 0
18
 1
 1
 0
19
 0
 0
 0
20
 1
 0
 1
21
 0
 1
 0
22
 1
 0
 0
23
 0
 0
 0
24
 1
 1
 0
25
 0
 0
 1
26
 1
 0
 0
27
 0
 1
 0
28
 1
 0
 0
29
 0
 0
 0
30
 1
 1
 1

Proportions:

Table E: Proportions of n values that each permutation occurs
n
 2
 3
 5
 proportion
 reduced proportion
1
 0
 0
 0
 8/30
 4/15
2
 1
 0
 0
 8/30
 4/15
3
 0
 1
 0
 4/30
 2/15
4
 1
 1
 0
 4/30
 2/15
5
 0
 0
 1
 2/30
 1/15
6
 1
 0
 1
 2/30
 1/15
7
 0
 1
 1
 1/30
 1/30
8
 1
 1
 1
 1/30
 1/30

For the range 1 to 30 for n:

000 occurs 8/30=4/15 of the time, so 4/15 of numbers don't have any of prime factors 2 or 3 or 5.
100 occurs 8/30=4/15 of the time, so 4/15 of numbers have prime factor 2 and don't have either of prime factors 3 or 5.
010 occurs 4/30=2/15 of the time, so 2/15 of numbers have prime factor 3 and don't have either of prime factors 2 or 5.
110 occurs 4/30=2/15 of the time, so 2/15 of numbers have prime factor 2 and 3 and don't have prime factors 5.
001 occurs 2/30=1/15 of the time, so 1/15 of numbers have prime factor 5 and don't have either of prime factors 2 or 3.
101 occurs 2/30=1/15 of the time, so 1/15 of numbers have prime factor 2 and 5 and don't have prime factors 3.
011 occurs 1/30 of the time.  So 1/30 of numbers have prime factor 3 and 5 and don't have prime factors 2.
111 occurs 1/30 of the time.  So 1/30 of numbers have prime factor 2 and 3 and 5.

These proportions sum to 1, corresponding to all natural numbers.

4/15+4/15+2/15+2/15+1/15+1/15+1/30+1/30=1. 

There are 4 distinct proportions.

{4/15, 2/15, 1/15, 1/30}

Each of the 4 proportions corresponds to 2 distinct prime factor permutations each, ie 4/15 corresponds to the {0,0,0} and {1,0,0} prime factor permutations.

Sequences:

The n values where these permutations occur give the eight sequences: (only first 20 values shown)

000: n: {1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73}
100: n: {2, 4, 8, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 52, 56, 58, 62, 64, 68, 74}
010: n: {3, 9, 21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 147}
110: n: {6, 12, 18, 24, 36, 42, 48, 54, 66, 72, 78, 84, 96, 102, 108, 114, 126, 132, 138, 144}
001: n: {5, 25, 35, 55, 65, 85, 95, 115, 125, 145, 155, 175, 185, 205, 215, 235, 245, 265, 275, 295}
101: n: {10, 20, 40, 50, 70, 80, 100, 110, 130, 140, 160, 170, 190, 200, 220, 230, 250, 260, 280, 290}
011: n: {15, 45, 75, 105, 135, 165, 195, 225, 255, 285, 315, 345, 375, 405, 435, 465, 495, 525, 555, 585}
111: n: {30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600}

First three values in each sequence:
{{1, 7, 11}, {2, 4, 8}, {3, 9, 21}, {6, 12, 18}, {5, 25, 35}, {10, 20, 40}, {15, 45, 75}, {30, 60, 90}}

Difference Sequences:

The difference sequences of the above:

First differences sequences:

000: Differences1[n]: {6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2}
100: Differences1[n]: {2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6}
010: Differences1[n]: {6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6}
110: Differences1[n]: {6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6}
001: Differences1[n]: {20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20}
101: Differences1[n]: {10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10, 20, 10}
011: Differences1[n]: {30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30}
111: Differences1[n]: {30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30}

Second differences sequences:

000: Abs[Differences2[n]]: {2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2}
100: Abs[Differences2[n]]: {2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2}
110: Abs[Differences2[n]]: {0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0}
010: Abs[Differences2[n]]: {6, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 6, 6}
001: Abs[Differences2[n]]: {10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10}
101: Abs[Differences2[n]]: {10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10}

Third differences sequences:

000: Abs[Differences3[n]]: {0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0}
100: Abs[Differences3[n]]: {0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0}
110: Abs[Differences3[n]]: {0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0}
010: Abs[Differences3[n]]: {0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0}

Fourth differences sequences:

000: Abs[Differences4[n]]: {0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 2}
100: Abs[Differences4[n]]: {2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0}
110: Abs[Differences4[n]]: {6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6}
010: Abs[Differences4[n]]: {6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6}

Fifth differences sequences:

000: Abs[Differences5[n]]: {0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0}
100: Abs[Differences5[n]]: {0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0}

Sixth differences sequences:

000: Abs[Differences6[n]]: {0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0}
100: Abs[Differences6[n]]: {0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0}

Seventh differences sequences:

000: Abs[Differences7[n]]: {0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0}
100: Abs[Differences7[n]]: {0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0}

Eighth differences sequences:

000: Abs[Differences8[n]]: {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}
100: Abs[Differences8[n]]: {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}

The count of how many difference sequences are required to give a sequence of all the same absolute value numbers:
Ie 4 difference sequences are required for 010:
010: n: {3, 9, 21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 147}
010: Differences1[n]: {6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6, 6, 6, 12, 6}
010: Abs[Differences2[n]]: {6, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 6, 6}
010: Abs[Differences3[n]]: {0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0}
010: Abs[Differences4[n]]: {6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6}

Relate these difference sequence counts to the proportions for each sequence, also check the actual difference values at each difference level, ie 2,6,10,30 for difference levels 8,4,2,1?
In this case, the differences sequences count is equal to the numerator of the proportion in table F.

Table F: Table E with differences sequences count column added
n
 2
 3
 5
 proportion
 reduced proportion
 differences sequences count
1
 0
 0
 0
 8/30
 4/15
 8
2
 1
 0
 0
 8/30
 4/15
 8
3
 0
 1
 0
 4/30
 2/15
 4
4
 1
 1
 0
 4/30
 2/15
 4
5
 0
 0
 1
 2/30
 1/15
 2
6
 1
 0
 1
 2/30
 1/15
 2
7
 0
 1
 1
 1/30
 1/30
 1
8
 1
 1
 1
 1/30
 1/30
 1


Add:

1. equivalent sequences using numerators and tuples etc
2. A309497 related transformed sequences, with example for constraint solver etc
3. add info on sequence 1, 1, 2, 4, 8, 16, 26, 50, 88, 176, 352, 576, 824, 1248, 2040, 4080, 8160, 16320, 21728, 34480, 53520, 69648, 111808, 223616, 326464 (distinct proportions)
4. simplified fractions with equivalent values (ie for larger permutation patterns)
5. add two more pages, ie for 2,3,5,7 and 2,3,5,7,11 (don't add the full 210 and 2310 tables, just the distinct 16 and 26 row tables)

Related Mathematica code:

add code for tuples, and other code to generate sequences etc snippets

(link to related code from here)