Re: who has the most divisors ??

I was originally trying to find the odds for tossing a counter onto a restricted times table =<n. I needed to find all the expressions of each number over the total numbers on the table.Please let me Know if this was of interest, i wish i had known this when I was in school instead of scribbling down on my 4ft (triangular shaped to save some space) list of all the unique divisors. (It Was fun though and still is great fun in my spare time away from work, to create colour coded charts of unique
fractions & (n*1/n)=(n/n)... 1ftdeep. Even if I'd known i think... no I know, i would have done it anyway.
Primality/Composite Test (D.E.2).

0, 0, 0, 0, 0, 0, 0, 0, 0, 1
0, 0, 0, 0, 0, 0, 0, 0, 1, 1
0, 0, 0, 0, 0, 0, 0, 1, 1, 1
0, 0, 0, 0, 0, 0, 1, 1, 1, 1
0, 0, 0, 0, 0, 1, 1, 1, 1, 1
0, 0, 0, 0, 1, 1, 1, 1, 2, 2
0, 0, 0, 1, 1, 1, 2, 2, 2, 3
0, 0, 1, 1, 2, 2, 3, 3, 4, 4
0, 1, 2, 3, 4, 5, 6, 7, 8, 9

0, 1, 3, 5, 8, 10,14,16....
The greatest difference between two(X & X-1) columns total values = Greatest divisor.

If we add together all Trunc (n/n-X) values (n/n-0 + n/n-1 +n/n-2) we can get the sum of all divisor expressions (Unique factors) that have been laid before.*e.g. (1)
If we see the remainders as a positional value of how far from the last actual divisor that appeared in this sequence is. e.g. 15/7 = 2 & 1/7 *e.g.
(2).
On this journey we obviously would have come across the divisors, so by restricting it at it's ClosestKnownSqrt, we only allow part of the function through we simplify it exponentionaly.

Each Factorial Degree
Ignore the remainder's, or take them from the original, Truncate... or all sort("Occam's Razor". we'll ignore them for we know there importance)

*E.g.. 1
r0 x/2 x/3 x/4 (00r01)...etc
01r00 00r1/2 00r1/3 00r1/4... etc
02r00 01r00
03r00 01r01 01r00
04r00 02r00 01r01 01r00
05r00 02r01 01r02 01r01 01r00
06r00 03r00 02r00 01r02 01r01 01r00
07r00 03r01 02r01 01r03 01r02 01r01 01r00
08r00 04r00 02r02 02r00 01r03 01r02 01r01 01r00
09r00 04r01 - so 4 was the last time it could divide and so is the sum of all
previous *2divisors i.e.. we take n-1 away(n+1, n*?) the difference is
the nth value of divisors (...)

*E.g.. 2
(Closest Known) Divisors Root's
How far from being a rectangle or square they are.
16 (16/16=1)
15 (16/15=1& 1 of 15 away)
14 (16/14=1& 2 of 14 away)
13 (16/13=1& 3 of 13 away)
12 (16/12=1& 4 of 12 away)
11 (16/11=1& 5 of 11 away)
10 (16/11=1& 6 of 10 away )
09 (16/9=1 & 7 of 9 away)
08 16 (16/8=2)
07 14 (16/7=2 & 2 of 7 away)
06 12 (16/6=2 & 4 of 6 away)
05 10 15 (16/5=3 & 1 of 5 away)
04 08 12 16 (16/4=4)
03 06 09 12 15 (16/3=5 & 1 of 3rd away)
02 04 06 08 10 12 14 16 (16/2=8)
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 (16/1=16)

we can simplify it further by restricting the equation to its "Closest
Known Sqrt" by truncating it's value again. This raises a further issue
closely linked with the triangular number function and therefore can be
left to the end. n/n is 0 out, n/n-1 is 1 out, n/n-2 is two out.. etc.
Instead of spreading the calculation and complicating its form, we can
use Tn(X^2/2 + X/2) but with a twist. We need 0+1+2... so we simply by
adding -1 to eliminate naught, but still keep true to form when we
calculate from our two point, 0:0 = 0, 1:0 = 1, 2:1 = 2, 3:2 = 3...

SUM Of n's Divisor Expression's in (X)Y=z (Two character space) ( DEZ's/DE2's, DE1's are 1 dimensional, DE3's I'll give an example at the bottom )

Trunc (^2root of n) = X
(Trunc (n/(X-0) = 0Y > Naught) + (Trunc (n/(X-1) = 1Y > Naught)... = Z
All - (X-1)^2+X/2 = Z-(X-1)Tn (Triangular number)

Then if n = 15
X=3
0y=5
1Y=7
2y=15
3ySTOP=0
(X-1)Tn=3 ANSWER=24

And then take n-1 away
Then if n-1 = 14
X=3
0y=4
1Y=7
2y=14
3ySTOP=0
(-1X)Tn=3 ANSWER=22

No. of DE2 Divisors=2

DE2 = Lowest Possible amount of answers from (X)Y to form a
distinguishable characteristics of primes and the prime square(closest
square, -1 divisor away) and other composites and there related squares
one divisor value below.

06 14 24 36 50 66 = 0+6+8+10.............
05 12 21 32 45 60 = 0+5+7+9+11.........
04 10 18 28 40 54 = 0+4+6+8+10.........
03 08 15 24 35 48 = 0+3+5+7+9+11.....
02 06 12 20 30 42 = 0+2+4+6+8+10.....
01 04 09 16 25 36 = 0+1+3+5+7+9+11.

Divisor Expressions of 2 multiple functions (X)Y = Z

(Decimal Expression of Z).

(X~first..Y~second...etc=Z)

n ODDS (No. Chance - (CKRoot2=<12) 600)

ONE and TWO Unique Divisors (ONE DE2 Divisors) {110/600}
/1/ 2, 3, 5, 7, 11, 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, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599

THREE and FOUR Unique Divisors (TWO DE2 Divisors) {190/600}
/4/ 6, 8, /9/ 10, 14, 15, 21, 22, /25/ 26, 27, 33, 34, 35, 38, 39, 46, /49/ 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, /121/ 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, /169/ 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218, 219, 221, 226, 235, 237, 247, 249, 253, 254, 259, 262, 265, 267, 274, 278, 287, /289/ 291, 295, 298, 299, 301, 302, 303, 305, 309, 314, 319, 321, 323, 326, 327, 329, 334, 335, 339, 341, 343, 346, 355, 358, /361/ 362, 365, 371, 377, 381, 382, 386, 391, 393, 394, 395, 398, 403, 407, 411, 413, 415, 417, 422, 427, 437, 445, 446, 447, 451, 453, 454, 458, 466, 469, 471, 473, 478, 481, 482, 485, 489, 493, 497, 501, 502, 505, 511, 514, 515, 517, 519, 526, 527, /529/ 533, 535, 537, 538, 542, 543, 545, 551, 553, 554, 559, 562, 565, 566, 573, 579, 581, 583, 586, 589, 591, 597

FIVE and SIX Unique Divisors (THREE DE2 Divisors) {72/600}
12, /16/ 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, /81/ 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 507, 508, 524, 531, 539, 548, 549, 556, 575, 578, 596

SEVEN and EIGHT Unique Divisors (FOUR DE2 Divisors) {103/600}
24, 30, 40, 42, 54, 56, /64/ 66, 70, 78, 88, 102, 104, 105, 110, 114, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 258, 266, 273, 282, 285, 286, 290, 296, 297, 310, 318, 322, 328, 344, 345, 351, 354, 357, 366, 370, 374, 375, 376, 385, 399, 402, 406, 410, 418, 424, 426, 429, 430, 434, 435, 438, 442, 455, 459, 465, 470, 472, 474, 483, 488, 494, 498, 506, 513, 518, 530, 534, 536, 555, 561, 568, 574, 582, 584, 590, 595, 598

NINE and TEN Unique Divisors (FIVE DE2 Divisors) {22/600}
/36/ 48, 80, /100/ 112, 162, 176, /196/ 208, /225//256/ 272, 304, 368, 405, /441/ 464, /484/ 496, /512/ 567, 592

ELEVEN and TWELVE Unique Divisors (SIX DE2 Divisors) {55/600}
60, 72, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 198, 200, 204, 220, 224, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 352, 364, 372, 380, 392, 414, 416, 444, 460, 476, 486, 490, 492, 495, 500, 516, 522, 525, 532, 544, 550, 558, 564, 572, 580, 585

THIRTEEN and FOURTEEN Unique Divisors (SEVEN DE2 Divisors) {3/600}
192, 320, 448

FIFTEEN and SIXTEEN Unique Divisors (EIGHT DE2 Divisors) {25/600}
120, /144/ 168, 210, 216, 264, 270, 280, 312, /324/ 330, 378, 384, 390, /400/ 408, 440, 456, 462, 510, 520, 546, 552, 570, 594

SEVENTEEN and EIGHTEEN Unique Divisors (NINE DE2 Divisors) {8/600}
180, 252, 288, 300, 396, 450, 468, 588

NINETEEN and TWENTY Unique Divisors (TEN DE2 Divisors) {5/600}
240, 336, 432, 528, 560

TWENTY-ONE and TWENTY-TWO Unique Divisors (ELEVEN DE2 Divisors) {1/600}
/576/

TWENTY-THREE and TWENTY-FOUR Unique Divisors (TWELVE DE2 Divisors) {6/600}
360, 420, 480, 504, 540, 600

etc...

Derren .

*~ 3D example (A)(B)C=Z.
This was originally for odds of landing on a Z on a restricted times table, equal to or fewer than n. Number One must be a square example of a prime as it has the same function in the cube. We are going to need a 3D projection of our times table cube, and a stone that defies gravity and can bounce within the confides of said cube for the later experiment but I'll show you any way.

01

04 08 xx xx
02 xx xx xx

09 18 27
06 12
03

xx 32 48 64
xx 24 36
xx 16
xx

25 50 75 100 125
xx 40 60 80
15 30 45
10 20
05

My drawings with colours and giving it the 3D illusion by raising the immediate right column up one..
The xx represents a variable(similar in value but not form) of these 3D rectangles added together in the repeated column below.
Repeated thus:

x 3 3 3 1
3 3 1 x 6 6 3
3 1 6 3 x 6 3
1 3 3 x 3

1(1) +7(8) +19(27) +37(64)
Three's on the out side, six's in the middle and leading to the Cube Prime 1 (Similar in the simplified function) at the top.
.

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