page4

Permutations with five variables: (mathematica code link1, simplify this code)


Permutation table C for x=5 binary variables has 2^x=2^5=32 permutations, ie: 00000,00001,00010,00011,00100,00101,00110,00111,01000,01001,01010,01011,01100,01101,01110,01111,10000,10001,10010,10011,10100,10101,10110,10111,11000,11001,11010,11011,11100,11101,11110,11111.

Table C: binary table for five variables
n
 variable1
 variable2
 variable3
 variable4
 variable5
1
 0
 0
 0
 0
 0
2
 0
 0
 0
 0
 1
3
 0
 0
 0
 1
 0
4
 0
 0
 0
 1
 1
5
 0
 0
 1
 0
 0
6
 0
 0
 1
 0
 1
7
 0
 0
 1
 1
 0
8
 0
 0
 1
 1
 1
9
 0
 1
 0
 0
 0
10
 0
 1
 0
 0
 1
11
 0
 1
 0
 1
 0
12
 0
 1
 0
 1
 1
13
 0
 1
 1
 0
 0
14
 0
 1
 1
 0
 1
15
 0
 1
 1
 1
 0
16
 0
 1
 1
 1
 1
17
 1
 0
 0
 0
 0
18
 1
 0
 0
 0
 1
19
 1
 0
 0
 1
 0
20
 1
 0
 0
 1
 1
21
 1
 0
 1
 0
 0
22
 1
 0
 1
 0
 1
23
 1
 0
 1
 1
 0
24
 1
 0
 1
 1
 1
25
 1
 1
 0
 0
 0
26
 1
 1
 0
 0
 1
27
 1
 1
 0
 1
 0
28
 1
 1
 0
 1
 1
29
 1
 1
 1
 0
 0
30
 1
 1
 1
 0
 1
31
 1
 1
 1
 1
 0
32
 1
 1
 1
 1
 1

Proportions:

For the range 1 to 2^x for x=5, for n:
00000 occurs 1/(2^x)=1/32 of the time
00001 occurs 1/32 of the time
00010 occurs 1/32 of the time
00011 occurs 1/32 of the time
00100 occurs 1/32 of the time
00101 occurs 1/32 of the time
00110 occurs 1/32 of the time
00111 occurs 1/32 of the time
01000 occurs 1/32 of the time
01001 occurs 1/32 of the time
01010 occurs 1/32 of the time
01011 occurs 1/32 of the time
01100 occurs 1/32 of the time
01101 occurs 1/32 of the time
01110 occurs 1/32 of the time
01111 occurs 1/32 of the time
10000 occurs 1/32 of the time
10001 occurs 1/32 of the time
10010 occurs 1/32 of the time
10011 occurs 1/32 of the time
10100 occurs 1/32 of the time
10101 occurs 1/32 of the time
10110 occurs 1/32 of the time
10111 occurs 1/32 of the time
11000 occurs 1/32 of the time
11001 occurs 1/32 of the time
11010 occurs 1/32 of the time
11011 occurs 1/32 of the time
11100 occurs 1/32 of the time
11101 occurs 1/32 of the time
11110 occurs 1/32 of the time
11111 occurs 1/32 of the time

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

1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32+1/32=1. 

Sequences:

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

00000: n: {1, 33, 65, 97}
00001: n: {2, 34, 66, 98}
00010: n: {3, 35, 67, 99}
00011: n: {4, 36, 68, 100}
00100: n: {5, 37, 69, 101}
00101: n: {6, 38, 70, 102}
00110: n: {7, 39, 71, 103}
00111: n: {8, 40, 72, 104}
01000: n: {9, 41, 73, 105}
01001: n: {10, 42, 74, 106}
01010: n: {11, 43, 75, 107}
01011: n: {12, 44, 76, 108}
01100: n: {13, 45, 77, 109}
01101: n: {14, 46, 78, 110}
01110: n: {15, 47, 79, 111}
01111: n: {16, 48, 80, 112}
10000: n: {17, 49, 81, 113}
10001: n: {18, 50, 82, 114}
10010: n: {19, 51, 83, 115}
10011: n: {20, 52, 84, 116}
10100: n: {21, 53, 85, 117}
10101: n: {22, 54, 86, 118}
10110: n: {23, 55, 87, 119}
10111: n: {24, 56, 88, 120}
11000: n: {25, 57, 89, 121}
11001: n: {26, 58, 90, 122}
11010: n: {27, 59, 91, 123}
11011: n: {28, 60, 92, 124}
11100: n: {29, 61, 93, 125}
11101: n: {30, 62, 94, 126}
11110: n: {31, 63, 95, 127}
11111: n: {32, 64, 96, 128}

The difference sequences of the above:
00000: Differences[n]: {32,32,32}
00001: Differences[n]: {32,32,32}
00010: Differences[n]: {32,32,32}
00011: Differences[n]: {32,32,32}
00100: Differences[n]: {32,32,32}
00101: Differences[n]: {32,32,32}
00110: Differences[n]: {32,32,32}
00111: Differences[n]: {32,32,32}
01000: Differences[n]: {32,32,32}
01001: Differences[n]: {32,32,32}
01010: Differences[n]: {32,32,32}
01011: Differences[n]: {32,32,32}
01100: Differences[n]: {32,32,32}
01101: Differences[n]: {32,32,32}
01110: Differences[n]: {32,32,32}
01111: Differences[n]: {32,32,32}
10000: Differences[n]: {32,32,32}
10001: Differences[n]: {32,32,32}
10010: Differences[n]: {32,32,32}
10011: Differences[n]: {32,32,32}
10100: Differences[n]: {32,32,32}
10101: Differences[n]: {32,32,32}
10110: Differences[n]: {32,32,32}
10111: Differences[n]: {32,32,32}
11000: Differences[n]: {32,32,32}
11001: Differences[n]: {32,32,32}
11010: Differences[n]: {32,32,32}
11011: Differences[n]: {32,32,32}
11100: Differences[n]: {32,32,32}
11101: Differences[n]: {32,32,32}
11110: Differences[n]: {32,32,32}
11111: Differences[n]: {32,32,32}


Permutation table D for x=5 prime divisors has maximum 2^x=2^5=32 distinct permutations, ie
for five prime divisors (2, 3, 5, 7, 11) the divisibility pattern:
00000,00001,00010,00011,00100,00101,00110,00111,01000,01001,01010,01011,01100,01101,01110,01111,10000,10001,10010,10011,10100,10101,10110,10111,11000,11001,11010,11011,11100,11101,11110,11111
gives the 32 distinct permutations in order of first appearance in Table D: 00000,10000,01000,00100,11000,00010,1010,1001,0110,0101,1110,0011,1101,1011,0111,1111.
The 32 first appearances occur at{1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310} n: .  These are the divisors of 2310.
Ie. in Table D, for n=3, 3 is divisible by 3 and not 2 and not 5 and not 7 and not 11, so the value of the four columns are 0,1,0,0,0 for row n=3.

---idea to check then erase:
(test algorithm for translation from four prime divisors of first occurrence to five prime divisors first occurrence (ie divisors of 210 and divisors of 2310 translation of binary sequences), see about creating a sequence that
describes where the new 0 or 1 is added  to create the five prime divisors sequence from the four prime divisors sequence etc..
four prime divisors: 0000,1000,0100,0010,1100,0001,1010,1001,0110,0101,1110,0011,1101,1011,0111,1111.
five prime divisors:
(four prime divisors sequence from: "https://jmorken.github.io/20210505%20tree%20of%20composites/tree%20of%20composites%20page3.html"
---end of idea to check then erase.

Table D: Prime Factor table for 5 prime factors (first 15 and last 15 rows of the total 2310 row table)
first 15 rows of the table:
n
 prime2
 prime3
 prime5
 prime7
 prime11
1
 0
 0
 0
 0
 0
2
 1
 0
 0
 0
 0
3
 0
 1
 0
 0
 0
4
 1
 0
 0
 0
 0
5
 0
 0
 1
 0
 0
6
 1
 1
 0
 0
 0
7
 0
 0
 0
 1
 0
8
 1
 0
 0
 0
 0
9
 0
 1
 0
 0
 0
10
 1
 0
 1
 0
 0
11
 0
 0
 0
 0
 1
12
 1
 1
 0
 0
 0
13
 0
 0
 0
 0
 0
14
 1
 0
 0
 1
 0
15
 0
 1
 1
 0
 0

(skipped middle 2310-30=2280 rows of the table)
last 15 rows of the table:
n
 prime2
 prime3
 prime5
 prime7
 prime11
2296
 1
 0
 0
 1
 0
2297
 0
 0
 0
 0
 0
2298
 1
 1
 0
 0
 0
2299
 0
 0
 0
 0
 1
2300
 1
 0
 1
 0
 0
2301
 0
 1
 0
 0
 0
2302
 1
 0
 0
 0
 0
2303
 0
 0
 0
 1
 0
2304
 1
 1
 0
 0
 0
2305
 0
 0
 1
 0
 0
2306
 1
 0
 0
 0
 0
2307
 0
 1
 0
 0
 0
2308
 1
 0
 0
 0
 0
2309
 0
 0
 0
 0
 0
2310
 1
 1
 1
 1
 1

Proportions:

Table E: Proportions of n values that each permutation occurs sorted descending by proportion.
n
 2
 3
 5
 7
 11
 sequence first three terms
 proportion
 reduced proportion
1
 0
 0
 0
 0
 0
 1, 13, 17 480/2310
 16/77
2
 1
 0
 0
 0
 0
 2, 4, 8 480/2310
 16/77
3
 0
 1
 0
 0
 0
 3, 9, 27 240/2310
 8/77
4
 1
 1
 0
 0
 0
 6, 12, 18 240/2310
 8/77
5
 0
 0
 1
 0
 0
 5, 25, 65 120/2310
 4/77
6
 1
 0
 1
 0
 0
 10, 20, 40 120/2310
 4/77
7
 0
 0
 0
 1
 0
 7, 49, 91 80/2310
 8/231
8
 1
 0
 0
 1
 0
 14, 28, 56 80/2310
 8/231
9
 0
 1
 1
 0
 0
 15, 45, 75 60/2310
 2/77
10
 1
 1
 1
 0
 0
 30, 60, 90 60/2310
 2/77
11
 0
 0
 0
 0
 1
 11, 121, 143 48/2310
 8/385
12
 1
 0
 0
 0
 1
 22, 44, 88 48/2310
 8/385
13
 0
 1
 0
 1
 0
 21, 63, 147
 40/2310
 4/231
14
 1
 1
 0
 1
 0
 42, 84, 126
 40/2310
 4/231
15
 0
 1
 0
 0
 1
 33, 99, 297 24/2310
 4/385
16
 1
 1
 0
 0
 1
 66, 132, 198 24/2310
 4/385
17
 0
 0
 1
 1
 0
 35, 175, 245
 20/2310 2/231
18
 1
 0
 1
 1
 0
 70, 140, 280
 20/2310 2/231
19
 0
 0
 1
 0
 1
 55, 275, 605
 12/2310 2/385
20
 1
 0
 1
 0
 1
 110, 220, 440
 12/2310 2/385
21
 0
 1
 1
 1
 0
 105, 315, 525
 10/2310 1/231
22
 1
 1
 1
 1
 0
 210, 420, 630
 10/2310 1/231
23
 0
 0
 0
 1
 1
 77, 539, 847
 8/2310 4/1155
24
 1
 0
 0
 1
 1
 154, 308, 616
 8/2310 4/1155
25
 0
 1
 1
 0
 1
 165, 495, 825
 6/2310 1/385
26
 1
 1
 1
 0
 1
 330, 660, 990
 6/2310 1/385
27
 0
 1
 0
 1
 1
 231, 693, 1617
 4/2310 2/1155
28
 1
 1
 0
 1
 1
 462, 924, 1386
 4/2310 2/1155
29
 0
 0
 1
 1
 1
 385, 1925, 2695
 2/2310 1/1155
30
 1
 0
 1
 1
 1
 770, 1540, 3080
 2/2310 1/1155
31
 0
 1
 1
 1
 1
 1155, 3465, 5775
 1/2310 1/2310
32
 1
 1
 1
 1
 1
 2310, 4620, 6930
 1/2310 1/2310


For the range 1 to 2310 for n:

00000 occurs 480/2310=16/77, so 16/77 of numbers don't have any of prime factors 2 or 3 or 5 or 7 or 11.
10000 occurs 480/2310=16/77, so 16/77 of numbers have prime factor 2 and don't have any of prime factors 3 or 5 or 7 or 11.
01000 occurs 240/2310=8/77, so 8/77 of numbers have prime factor 3 and don't have any of prime factors 2 or 5 or 7 or 11.
11000 occurs 240/2310=8/77, so 8/77 of numbers have prime factor 2 and 3 and don't have any of prime factors 5 or 7 or 11.
00100 occurs 120/2310=4/77, so 4/77 of numbers have prime factor 5 and don't have any of prime factors 2 or 3 or 7 or 11.
10100 occurs 120/2310=4/77, so 4/77 of numbers have prime factor 2 and 5 and don't have any of prime factors 3 or 7 or 11.
00010 occurs 80/2310=8/231, so 8/231 of numbers have prime factor 7 and don't have any of prime factors 2 or 3 or 5 or 11.
10010 occurs 80/2310=8/231, so 8/231 of numbers have prime factor 2 and 7 and don't have any of prime factors 3 or 5 or 11.
01100 occurs 60/2310=2/77, so 2/77 of numbers have prime factor 3 and 5 and don't have any of prime factors 2 or 7 or 11.
11100 occurs 60/2310=2/77, so 2/77 of numbers have prime factor 2 and 3 and 5 and don't have any of prime factors 7 or 11.
00001 occurs 48/2310=8/385, so 8/385 of numbers have prime factor 11 and don't have any of prime factors 2 or 3 or 5 or 7.
10001 occurs 48/2310=8/385, so 8/385 of numbers have prime factor 2 and 11 and don't have any of prime factors 3 or 5 or 7.
01010 occurs 40/2310=4/231, so 4/231 of numbers have prime factor 3 and 7 and don't have any of prime factors 2 or 5 or 11.
11010 occurs 40/2310=4/231, so 4/231 of numbers have prime factor 2 and 3 and 7 and don't have any of prime factors 5 or 11.
01001 occurs 24/2310=4/385, so 4/385 of numbers have prime factor 3 and 11 and don't have any of prime factors 2 or 5 or 7.
11001 occurs 24/2310=4/385, so 4/385 of numbers have prime factor 2 and 3 and 11 and don't have any of prime factors 5 or 7.
00110 occurs 20/2310=2/231, so 2/231 of numbers have prime factor 5 and 7 and don't have any of prime factors 2 or 3 or 11.
10110 occurs 20/2310=2/231, so 2/231 of numbers have prime factor 2 and 5 and 7 and don't have any of prime factors 3 or 11.
00101 occurs 12/2310=2/385, so 2/385 of numbers have prime factor 5 and 11 and don't have any of prime factors 2 or 3 or 7.
10101 occurs 12/2310=2/385, so 2/385 of numbers have prime factor 2 and 5 and 11 and don't have any of prime factors 3 or 7.
01110 occurs 10/2310=1/231, so 1/231 of numbers have prime factor 3 and 5 and 7 and don't have any of prime factors 2 or 11.
11110 occurs 10/2310=1/231, so 1/231 of numbers have prime factor 2 and 3 and 5 and 7 and don't have prime factor 11.
00011 occurs 8/2310=4/1155, so 4/1155 of numbers have prime factor 7 and 11 and don't have any of prime factors 2 or 3 or 5.
10011 occurs 8/2310=4/1155, so 4/1155 of numbers have prime factor 2 and 7 and 11 and don't have any of prime factors 3 or 5.
01101 occurs 6/2310=1/385, so 1/385 of numbers have prime factor 3 and 5 and 11 and don't have any of prime factors 2 or 7.
11101 occurs 6/2310=1/385, so 1/385 of numbers have prime factor 2 and 3 and 5 and 11 and don't have prime factor 7.
01011 occurs 4/2310=2/1155, so 2/1155 of numbers have prime factor 3 and 7 and 11 and don't have any of prime factors 2 or 5.
11011 occurs 4/2310=2/1155, so 2/1155 of numbers have prime factor 2 and 3 and 7 and 11 and don't have prime factor 5.
00111 occurs 2/2310=1/1155, so 1/1155 of numbers have prime factor 5 and 7 and 11 and don't have any of prime factors 2 or 3.
10111 occurs 2/2310=1/1155, so 1/1155 of numbers have prime factor 2 and 5 and 7 and 11 and don't have prime factor 3.
01111 occurs 1/2310=1/2310, so 1/2310 of numbers have prime factor 3 and 5 and 7 and 11 and don't have prime factor 2.
11111 occurs 1/2310=1/2310, so 1/2310 of numbers have prime factor 2 and 3 and 5 and 7 and 11.

[additional proportions: (check these for larger primorials)]

edit all the below for 2310:

Distinct numerators on the non-reduced proportions: 1, 2, 4, 6, 8, 12, 24, 48.

The 32 proportions sum to 1, corresponding to all natural numbers.

16/77+16/77+8/77+8/77+4/77+4/77+8/231+8/231+2/77+2/77+8/385+8/385+4/231+4/231+4/385+4/385+2/231+2/231+2/385+2/385+1/231+1/231+4/1155+4/1155+1/385+1/385+2/1155+2/1155+1/1155+1/1155+1/2310+1/2310=1.

There are 16 distinct proportions.

The 16 distinct proportions sorted largest to smallest:
16/77, 8/77, 4/77, 8/231, 2/77, 8/385, 4/231, 4/385, 2/231, 2/385, 1/231, 4/1155, 1/385, 2/1155, 1/1155, 1/2310
The 16 distinct proportions sorted smallest to largest:
1/2310, 1/1155, 2/1155, 1/385, 4/1155, 1/231, 2/385, 2/231, 4/385, 4/231, 8/385, 2/77, 8/231, 4/77, 8/77, 16/77

Distinct numerators of the reduced proportions: 1, 2, 4, 8, 16.
Distinct denominators of the reduced proportions: 77, 231, 385, 1155, 2310.

Each of the 16 proportions corresponds to 2 distinct prime factor permutations each, ie 16/77 corresponds to the {0,0,0,0,0} and {1,0,0,0,0} prime factor permutations.

Sequences:

The n values where these permutations occur give the 32 sequences: (only first 20 values shown)
Each sequence starts with a distinct divisor of 210.  The sequences are sorted in descending order by the corresponding proportions.

0000: n: {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83}
1000: n: {2, 4, 8, 16, 22, 26, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86}

0100: n: {3, 9, 27, 33, 39, 51, 57, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 153, 159, 171}
1100: n: {6, 12, 18, 24, 36, 48, 54, 66, 72, 78, 96, 102, 108, 114, 132, 138, 144, 156, 162, 174}

0010: n: {5, 25, 55, 65, 85, 95, 115, 125, 145, 155, 185, 205, 215, 235, 265, 275, 295, 305, 325, 335}
1010: n: {10, 20, 40, 50, 80, 100, 110, 130, 160, 170, 190, 200, 220, 230, 250, 260, 290, 310, 320, 340}

0001: n: {7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511}
1001: n: {14, 28, 56, 98, 112, 154, 182, 196, 224, 238, 266, 308, 322, 364, 392, 406, 434, 448, 476, 518}

0110: n: {15, 45, 75, 135, 165, 195, 225, 255, 285, 345, 375, 405, 435, 465, 495, 555, 585, 615, 645, 675}
1110: n: {30, 60, 90, 120, 150, 180, 240, 270, 300, 330, 360, 390, 450, 480, 510, 540, 570, 600, 660, 690}

0101: n: {21, 63, 147, 189, 231, 273, 357, 399, 441, 483, 567, 609, 651, 693, 777, 819, 861, 903, 987, 1029}
1101: n: {42, 84, 126, 168, 252, 294, 336, 378, 462, 504, 546, 588, 672, 714, 756, 798, 882, 924, 966, 1008}

0011: n: {35, 175, 245, 385, 455, 595, 665, 805, 875, 1015, 1085, 1225, 1295, 1435, 1505, 1645, 1715, 1855, 1925, 2065}
1011: n: {70, 140, 280, 350, 490, 560, 700, 770, 910, 980, 1120, 1190, 1330, 1400, 1540, 1610, 1750, 1820, 1960, 2030}

0111: n: {105, 315, 525, 735, 945, 1155, 1365, 1575, 1785, 1995, 2205, 2415, 2625, 2835, 3045, 3255, 3465, 3675, 3885, 4095}
1111: n: {210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 2520, 2730, 2940, 3150, 3360, 3570, 3780, 3990, 4200}




First three values in each of the 32 sequences:

{{1,13,17}, {2,4,8}, {3,9,27}, {6,12,18}, {5,25,65}, {10,20,40}, {7,49,91}, {14,28,56}, {15,45,75}, {30,60,90}, {11,121,143}, {22,44,88}, {21,63,147}, {42,84,126}, {33,99,297}, {66,132,198}, {35,175,245}, {70,140,280}, {55,275,605}, {110,220,440}, {105,315,525}, {210,420,630}, {77,539,847}, {154,308,616}, {165,495,825}, {330,660,990}, {231,693,1617}, {462,924,1386}, {385,1925,2695}, {770,1540,3080}, {1155,3465,5775}, {2310,4620,6930}}

Difference Sequences:

First differences sequences for the first 20 values:

0000: Differences1[n]: {}
1000: Differences1[n]: {}
0100: Differences1[n]: {}
1100: Differences1[n]: {}
0010: Differences1[n]: {}
1010: Differences1[n]: {}
0001: Differences1[n]: {}
1001: Differences1[n]: {}
0110: Differences1[n]: {}
1110: Differences1[n]: {}
0101: Differences1[n]: {}
1101: Differences1[n]: {}
0011: Differences1[n]: {}
1011: Differences1[n]: {}
0111: Differences1[n]: {}
1111: Differences1[n]: {}

Sequence pattern lengths:

Ie. The repeating pattern length of 00000 and 10000 is 480 given by the numerator of the proportion 480/2310.

Difference sequences offsets: (add mathematica code link for difference sequences offsets based on the page3 linked code)

Ie: 00000 is the same sequence as 10000, with a 480/2=240 offset. (doublecheck)

Shifting the difference sequence y by z amount x gives
0000->1000 offset by 480/2=240. (double check)
0100->1100 offset by x (double check these offsets are also the non reduced proportion numerator/2.
0010->1010 offset by x
0001->1001 offset by x
0110->1110 offset by x
0101->1101 offset by x
0011->1011 offset by x
0111->1111 offset by x

For any length of the sequences where the last value is greater or equal to 2309, the below table gives the distinct values in the first differences sequences.
For example for sequence 00000: {1,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197,199,211,221,223,227,229,233,239,241,247,251,257,263,269,271,277,281,283,289,293,299,307,311,313,317,323,331,337,347,349,353,359,361,367,373,377,379,383,389,391,397,401,403,409,419,421,431,433,437,439,443,449,457,461,463,467,479,481,487,491,493,499,503,509,521,523,527,529,533,541,547,551,557,559,563,569,571,577,587,589,593,599,601,607,611,613,617,619,629,631,641,643,647,653,659,661,667,673,677,683,689,691,697,701,703,709,713,719,727,731,733,739,743,751,757,761,767,769,773,779,787,793,797,799,809,811,817,821,823,827,829,839,841,851,853,857,859,863,871,877,881,883,887,893,899,901,907,911,919,923,929,937,941,943,947,949,953,961,967,971,977,983,989,991,997,1003,1007,1009,1013,1019,1021,1027,1031,1033,1037,1039,1049,1051,1061,1063,1069,1073,1079,1081,1087,1091,1093,1097,1103,1109,1117,1121,1123,1129,1139,1147,1151,1153,1157,1159,1163,1171,1181,1187,1189,1193,1201,1207,1213,1217,1219,1223,1229,1231,1237,1241,1247,1249,1259,1261,1271,1273,1277,1279,1283,1289,1291,1297,1301,1303,1307,1313,1319,1321,1327,1333,1339,1343,1349,1357,1361,1363,1367,1369,1373,1381,1387,1391,1399,1403,1409,1411,1417,1423,1427,1429,1433,1439,1447,1451,1453,1457,1459,1469,1471,1481,1483,1487,1489,1493,1499,1501,1511,1513,1517,1523,1531,1537,1541,1543,1549,1553,1559,1567,1571,1577,1579,1583,1591,1597,1601,1607,1609,1613,1619,1621,1627,1633,1637,1643,1649,1651,1657,1663,1667,1669,1679,1681,1691,1693,1697,1699,1703,1709,1711,1717,1721,1723,1733,1739,1741,1747,1751,1753,1759,1763,1769,1777,1781,1783,1787,1789,1801,1807,1811,1817,1819,1823,1829,1831,1843,1847,1849,1853,1861,1867,1871,1873,1877,1879,1889,1891,1901,1907,1909,1913,1919,1921,1927,1931,1933,1937,1943,1949,1951,1957,1961,1963,1973,1979,1987,1993,1997,1999,2003,2011,2017,2021,2027,2029,2033,2039,2041,2047,2053,2059,2063,2069,2071,2077,2081,2083,2087,2089,2099,2111,2113,2117,2119,2129,2131,2137,2141,2143,2147,2153,2159,2161,2171,2173,2179,2183,2197,2201,2203,2207,2209,2213,2221,2227,2231,2237,2239,2243,2249,2251,2257,2263,2267,2269,2273,2279,2281,2287,2291,2293,2297,2309}
The sorted distinct values in the first difference sequence are: {2, 4, 6, 8, 10, 12, 14} and the count of distinct values is 7.

Table F: Table E with differences sequences count column added
n
 2
 3
 5
 7
 11
 sequence first three terms
 proportion
 reduced proportion
 first differences sequence distinct terms first differences sequence distinct terms counts
1
 0
 0
 0
 0
 0
 1, 13, 17 480/2310
 16/77

2
 1
 0
 0
 0
 0
 2, 4, 8 480/2310
 16/77

3
 0
 1
 0
 0
 0
 3, 9, 27 240/2310
 8/77

4
 1
 1
 0
 0
 0
 6, 12, 18 240/2310
 8/77

5
 0
 0
 1
 0
 0
 5, 25, 65 120/2310
 4/77

6
 1
 0
 1
 0
 0
 10, 20, 40 120/2310
 4/77

7
 0
 0
 0
 1
 0
 7, 49, 91 80/2310
 8/231

8
 1
 0
 0
 1
 0
 14, 28, 56 80/2310
 8/231

9
 0
 1
 1
 0
 0
 15, 45, 75 60/2310
 2/77

10
 1
 1
 1
 0
 0
 30, 60, 90 60/2310
 2/77

11
 0
 0
 0
 0
 1
 11, 121, 143 48/2310
 8/385

12
 1
 0
 0
 0
 1
 22, 44, 88 48/2310
 8/385

13
 0
 1
 0
 1
 0
 21, 63, 147
 40/2310
 4/231

14
 1
 1
 0
 1
 0
 42, 84, 126
 40/2310
 4/231

15
 0
 1
 0
 0
 1
 33, 99, 297 24/2310
 4/385

16
 1
 1
 0
 0
 1
 66, 132, 198 24/2310
 4/385

17
 0
 0
 1
 1
 0
 35, 175, 245
 20/2310 2/231

18
 1
 0
 1
 1
 0
 70, 140, 280
 20/2310 2/231

19
 0
 0
 1
 0
 1
 55, 275, 605
 12/2310 2/385

20
 1
 0
 1
 0
 1
 110, 220, 440
 12/2310 2/385

21
 0
 1
 1
 1
 0
 105, 315, 525
 10/2310 1/231

22
 1
 1
 1
 1
 0
 210, 420, 630
 10/2310 1/231

23
 0
 0
 0
 1
 1
 77, 539, 847
 8/2310 4/1155

24
 1
 0
 0
 1
 1
 154, 308, 616
 8/2310 4/1155

25
 0
 1
 1
 0
 1
 165, 495, 825
 6/2310 1/385

26
 1
 1
 1
 0
 1
 330, 660, 990
 6/2310 1/385

27
 0
 1
 0
 1
 1
 231, 693, 1617
 4/2310 2/1155

28
 1
 1
 0
 1
 1
 462, 924, 1386
 4/2310 2/1155

29
 0
 0
 1
 1
 1
 385, 1925, 2695
 2/2310 1/1155

30
 1
 0
 1
 1
 1
 770, 1540, 3080
 2/2310 1/1155

31
 0
 1
 1
 1
 1
 1155, 3465, 5775
 1/2310 1/2310

32
 1
 1
 1
 1
 1
 2310, 4620, 6930
 1/2310 1/2310


Values in column "first differences sequences values":
2,4,6,8,10,12,14,18,20,28,30,42,60,70,84,140,210. (double check)
(for differences sequence: {2, 2, 2, 2, 2, 2, 4, 2, 8, 2, 12, 18, 10, 14, 56, 70} 140-84=56 is the only value not in the original sequence, 56=48+8, 56/210=8/30, check more)
Total terms: 17. (double check)

Values in column "first differences distinct terms":
1,2,3,5.
Total terms: 4.
Counts of occurrences of each value {1,2,3,5} are {2,6,6,2}:
1->2, 2->6, 3->6, 5->2. (check these counts of occurrences more)


Pairwise difference sequences: (mathematica code link for pairwise differences) (change this link to new code for this page)

Ie: take differences of two sequences with the same proportions:
a0000 = {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83} (*48/210 proportion sequence*)
a1000 = {2, 4, 8, 16, 22, 26, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86} (*48/210 proportion sequence*)
Abs[a0000 - a1000] (*outputs: 1, 7, 5, 1, 3, 3, 3, 3, 1, 3, 3, 5, 5, 3, 3, 1, 3, 3, 3, 3*)
Tally[Sort[Abs[a0000 - a1000]]] (*outputs: {1, 4}, {3, 12}, {5, 3}, {7, 1}*)
Sort[DeleteDuplicates[Abs[a0000 - a1000]]] (*outputs: 1, 3, 5, 7*)
CountDistinct[Abs[a0000 - a1000]] (*outputs: 4*)

Take pairwise difference sequence for range up to 210 for the full patterns etc.

show pairwise difference sequences for the first 20 values here:
0000-1000: Differences1[n]: {1, 7, 5, 1, 3, 3, 3, 3, 1, 3, 3, 5, 5, 3, 3, 1, 3, 3, 3, 3}
0100-1100: Differences1[n]: {3, 3, 9, 9, 3, 3, 3, 3, 9, 9, 3, 3, 3, 3, 9, 9, 3, 3, 3, 3}
0010-1010: Differences1[n]: {5, 5, 15, 15, 5, 5, 5, 5, 15, 15, 5, 5, 5, 5, 15, 15, 5, 5, 5, 5}
0001-1001: Differences1[n]: {7, 21, 21, 7, 7, 21, 21, 7, 7, 21, 21, 7, 7, 21, 21, 7, 7, 21, 21, 7}
0110-1110: Differences1[n]: {15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15}
0101-1101: Differences1[n]: {21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21}
0011-1011: Differences1[n]: {35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35}
0111-1111: Differences1[n]: {105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105}

Table G: Pairwise differences table
n
 pairwise differences
 tally
 distinct terms distinct terms counts
1
 0000-1000
 {1, 4}, {3, 12}, {5, 3}, {7, 1} 1, 3, 5, 7 4
2
 0100-1100
 {3, 14}, {9, 6}
 3, 9
 2
3
 0010-1010
 {5, 14}, {15, 6}
 5, 15
 2
4
 0001-1001
 {7, 10}, {21, 10}
 7, 21
 2
5
 0110-1110
 {15, 20}
 15
 1
6
 0101-1101
 {21, 20}
 21
 1
7
 0011-1011
 {35, 20}
 35
 1
8
 0111-1111
 {105, 20}
 105
 1



Add:

1. equivalent sequences using numerators and tuples etc
2. A309497 related transformed sequences, with example for constraint solver etc, also for sequences with equal proportions check for correspondence in their A309497 related transform sequences (ie similar to the first and half reversing for prime factor 2 added, since those two sequences also have equal proportions).
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 2,3,5,7,11 and 2,3,5,7,11,13 (don't add the full 2310 row and 30030 row tables, just the distinct 16 and 26 row tables)
6. add seperate H-tree visualization for page1, page2, page3, ... etc combined to show fractions etc for multiple permutation levels.
Related Mathematica code:
7. take the product of all divisors of a primorial ie take the product of the divisors of 2310, (the set of the divisors ie 32 divisors, give the set of permutations of the prime factors, see if the product of them has a simplified or reduced prime factorization similar to how some proportions are equal even though their sequences start with different divisors etc..

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

(link to related code from here)