The left and right numbers are linked. What should the last number on the right be, and why?

\begin{align} 135759&: 1 \\ 151364&: 4 \\ 255075&: 9 \\ 279422&: 36 \\ 292620&: 91 \\ 348777&: 135 \\ 398067&: 147 \\ 417894&: 265 \\ 459431&: 279 \\ 478926&: 307 \\ 609941&: 363 \\ 689245&: 435 \\ 814576&:\space? \\ \end{align}

Before anyone bothers, no right is not simply a polynomial of left (of course we could certainly fit one, but there is another way).

Hint

There is information somewhere in the community wiki post that may prove useful

Hint 2

The mapping from left to right is many to one and there are infinite possibilities for right.

Two more entries are
\begin{align}0&: 1\\1&: 0\\\end{align}

Hint 3

Nothing special about these, they are really just some more examples...

Since there were none, here are some with $8$s on the right: \begin{align}157388&: 8\\272813&: 18\\276276&:28\\291384&: 88\\\end{align}
To have a cube $\gt1$ on both sides, left must have at least $7$ digits, here are a few: \begin{align}1259712&: 27\\224755712&: 729\\559476224&: 1000\\1427628376&: 6859\\\end{align}

• Is this sequence a sequence? As in, In order without gaps? Dots . . . after the "?" ? – humn May 12 '16 at 8:14
• @humn I could put them in any order if that is your question - the left and right numbers are logically linked, as stated. My first hint would have shown this to be true; oh well. – Jonathan Allan May 12 '16 at 10:16
• is it just me, or does 'logically' linked suggest bitwise operations for others as well? – elias May 12 '16 at 11:48
• Is the list potentially infinite? Is it complete other than the "?" ? Would that be telling? Would answering that be telling? – humn May 22 '16 at 6:18
• @humn answered with hint 2 – Jonathan Allan May 22 '16 at 7:14

What's the $\bf ?$ in $~ \bf 814576 : \, ? ~$ ? $\require{begingroup} \begingroup \let \SSS \tiny \let \SS \scriptsize \let \S \small \def \T {\small\sf} \def \Q #1{{ \bf ? { \S\raise2mu ( } #1 { \S\raise2mu ) } }} \def \R #1{ \phantom{\Q0} \llap{ \bf #1 \kern 7mu } } \def \x #1{{\SSS\kern1mu\raise2mu \times \S\kern5mu 3 \kern1mu\SS\raise7mu #1}} \def \p { \kern8mu{ \S \raise1mu + }\kern8mu} \def \D { \kern6em } \def \E #1{ \D \llap{ #1 \kern4mu { \S\raise1mu = } \kern6mu } }$

$\kern-12mu \E{ \bf ? } 514$

Essential $\bf 1 \over 10\raise-4mu{\small\,3}$ of the story

$\E{ \Q{\T digit} }$ how many enclosed regions, typically loops, in the drawing of the single $\T digit$
$\E{ \Q{\T digits} }$ decimal value of the ternary number formed
$\D \raise-6mu\strut$ when each $\T digit$ of $\T digits$ is replaced by $\Q{\T digit}$
$\E{ \Q{814576} } \Q 8 \x 5 \p\Q 1 \x 4 \p\Q 4 \x 3 \p\Q 5 \x 2 \p\Q 7 \x 1 \p\Q 6 \x 0$
$\E{} \R2 \x5 \p\R0 \x4 \p\R1 \x3 \p\R0 \x2 \p\R0 \x1 \p\R1 \x0$
$\E{} 201001\raise-4mu{\small\,3}$
$\E{} 514\raise-4mu{\scriptsize\,10}$ $\endgroup$

Middle $\bf 1 \over 10\raise-4mu{\small\,3}$ of the story

Notice the following retrieval from the community evidence locker. It originated from using various radices for the numbers being puzzled and displaying the only two radices with suggestive patterns.

    left  : right         binary        ternary
135759 : 1                  1              1
151364 : 4                1..             11
255075 : 9               1..1            1..
279422 : 36            1..1..           11..
292620 : 91           1.11.11          1.1.1
348777 : 135         1....111          12...
398067 : 147         1..1..11          1211.
417894 : 265        1....1..1         1..211
459431 : 279        1...1.111         1.11..
478926 : 307        1..11..11         1.21.1
609941 : 363        1.11.1.11         11111.
689245 : 435        11.11..11         121.1.


Watch what happens, starting with the dots, when ...

... the middle two columns are ignored and left-column digits without enclosures are dotted out.

  left                          right ternary
.....9                                     1
....64                                    11
...0..                                   1..
..94..                                  11..
.9.6.0                                 1.1.1
.48...                                 12...
.9806.                                 1211.
4..894                                1..211
4.94..                                1.11..
4.89.6                                1.21.1
60994.                                11111.
689.4.                                121.1.   
No wonder the ternaries had so few $\small\tt 2$s, a peculiarity that had been noted early without direct result. Only the digit $8$ has two loops and thus transforms into $\small\tt 2$, while each of four digits$-0 \, 4 \, 6 \, 9-$has just one loop (or enclosed region) and transforms into $\small\tt 1$.

And no wonder that a few ternary numbers were exactly as wide as the left-column numbers and that none were wider, which was not noted along the way.

Final $\bf 1 \over 10\raise-4mu{\small\,3}$ of the story

How would anyone notice the above relationship? By following up on clues, such as those just mentioned, and allowing for dumb luck coincidence.

The records room at Puzzling HQ has a dossier on The Case of the Really, really, really hard sequence, which was reported just 11 days after the present puzzle and hinges on an eerily comparable ternary modus operandi. Sure enough, that case was cracked by the poser of the present puzzle, in barely 86 minutes and 48 seconds. To see a new puzzle that uses a similar device must have been precious! And then to write a solution that flaunts the very key to their own unsolved puzzle.

The present puzzle's poser's pathological ternary obsession did not go unnoticed, nor uncontracted, by the detective assigned to this case. Sooner or later, well, much after the hint ...

... $0 \kern2mu {:} \kern1mu 1$ revealed that two wildly different numbers, $135759$ and $0$, can produce the same degenerate result, $1$, it was time to consider that individual digits of the puzzling numbers might be nullifying each other or even being ignored. The binary and ternary patterns above seemed like the simplest leads to follow when embarking on this. (Not) one of the (first) possibilities to touch on, if only to shake up thought patterns, was the graphical way of looking at digits in that other case.

And it happened to work. Tickle me pink.

• I wish I could +1 again for linking to the inspiration of this puzzle. – Anton Jun 17 '16 at 8:16
• Very impressive! – Gareth McCaughan Jun 17 '16 at 10:37
• Hooray. Sorry I was so lax with responding to my notification! – Jonathan Allan Jun 17 '16 at 12:00

(Community evidence locker)

All factors                  1 : 135759   45253  10443  3481  2301   767   177    59    39    13   3
2    4 : 151364   75682  37841  1916   958   479   316   158    79     4   2
3    9 : 255075   85025  51015 17005 13425 10203  4475  3401  2685  1425 895   537 475 285 179 95 75 57 25 19 15 5 3
2 3 4 6 12 9 18   36 : 279422  139711  25402 21494 12701 10747  1954   977   286   143  26    22  13  11   2
7 13   91 : 292620  146310  97540 73155 58524 48770 29262 24385 19508 14631 9754 4877  60  30  20  15 12 10  6  5  4 3 2
3 5 9 15 27 45  135 : 348777  116259  38753 31707 26829 10569  8943  3523  2981  2439 1287  813 429 271 143 117 99 39 33 13 11 9 3
3 7 21 49  147 : 398067  132689      3
5 53  265 : 417894  208947 139298 69649 24582 12291 8194 4097 1734 1446 867 723 578 482 289 241 102 51 34 17 6 3 2
3 9 31 93  279 : 459431   65633      7
307 : 478926  239463 159642 79821 68418 53214 34209 26607 22806 17738 11403 9774 8869 7602 4887 3801 3258 2646 2534 1629 1323 1267 1086 882 543 441 378 362 294 189 181 147 126 98 63 54 49 42 27 21 18 14 9 7 6 3 2
3 11 33 121  363 : 609941    1699    359
3 5 15 29 87 145  435 : 689245  137849      5
?  : 814576  407288 203644 116368 101822 58184 50911 29092 16624 14546 8312 7273 4156 2078 1039 784 392 196 112 98 56 49 28 16 14 8 7 4 2

Prime factors                1 : 135759         3               13    59 59
2 2   4 : 151364  2 2                               79   479
3 3         9 : 255075         3    5 5             19     179
3 3   2 2  36 : 279422   2                11   13               977
13   7                  91 : 292620  2 2    3     5                             4877
5  3 3 3      135 : 348777        3 3         11   13            271
7 7       3        147 : 398067         3                                    132689
53          5             265 : 417894   2     3                 17 17       241
31            3 3       279 : 459431                    7                         65633
307                        307 : 478926   2   3 3 3       7 7                181
11 11        3        363 : 609941                                        359 1699
29        5    3        435 : 689245               5                              137849
?  : 814576  2 2 2 2          7 7                     1039

ternary               binary    largest prime       largest prime     binary    ternary
factor                     factor
2.12               111.11         59    135759 : 1       1              1          1
1222.2            111.11111        479    151364 : 4       2             1.          2
2.122             1.11..11        179    255075 : 9       3             11         1.
11...12           1111.1...1        977    279422 : 36      3             11         1.
2.2..122        1..11....11.1       4877    292620 : 91      13          11.1        111
1.1..1            1....1111        271    348777 : 135     5            1.1         12
2.2.2...1.2   1......11..1.1...1     132689    398067 : 147     7            111         21
22221             1111...1        241    417894 : 265     53        11.1.1       1222
1.1.....212    1.........11....1      65633    459431 : 279     31         11111       1.11
2.2.1             1.11.1.1        181    478926 : 307     307    1..11..11     1.21.1
2.22221          11.1.1...11       1699    609941 : 363     11          1.11        1.2
21.....2112   1....11.1..1111..1     137849    689245 : 435     29         111.1       1..2
11.2111          1......1111       1039    814576 : ?

ternary                      binary                              binary        ternary
2.22..2..1.         1....1..1..1..1111       135759 : 1                  1              1
212..122..2          1..1..1111.1...1..       151364 : 4                1..             11
11.22122..2.         11111..1...11...11       255075 : 9               1..1            1..
112.12.21222        1...1....11.111111.       279422 : 36            1..1..           11..
1122121.121.        1...111.111....11..       292620 : 91           1.11.11          1.1.1
1222.11.22..        1.1.1.1..1..11.1..1       348777 : 135         1....111          12...
2.2.2...1.2.        11....1..1.1111..11       398067 : 147         1..1..11          1211.
21..2..2.12.        11..11......11..11.       417894 : 265        1....1..1         1..211
2121...12222        111......1.1.1..111       459431 : 279        1...1.111         1.11..
22..22222...        111.1..111.11..111.       478926 : 307        1..11..11         1.21.1
1.1.2222..1.2       1..1.1..111.1..1.1.1       609941 : 363        1.11.1.11         11111.
1.22...11.121       1.1.1....1...1.111.1       689245 : 435        11.11..11         121.1.
11121.11.1111       11...11.11.11111....       814576 : ?

L :from base 16   to base 10   R (result)
135759         1267545      1
151364         1381220      4
255075         2445429      9
279422         2593826      36
292620         2696736      91
348777         3442551      135
398067         3768423      147
417894         4290708      265
459431         4559921      279
478926         4688166      307
609941         6330689      363
689245         6853189      435
814576         8471926      ?

• 65633 and 132689 seem somewhat close to 2^16 and 2^17, but it could be not related... – ffao May 12 '16 at 6:13
• While we're talking powers of 2, I remark that the last LH number 814576 is, if we give it a leading zero 0814576, an anagram of 2^20=1048576. Almost certainly coincidence, I think. – Gareth McCaughan May 12 '16 at 12:13
• Surprisingly many prime factors squared in both columns. Surprisingly few 2s in the ternaries of the right column #s (even when disregarding the leading digits of the largest numbers). – humn May 15 '16 at 20:44
• @gannolloy If you edit it it updates. I don't think it causes it to jump up the "active" tab in the questions section though. From experience the puzzle setter (me) also does not get a notification - and does not get notified of comments that are not addressed directly to them (like I was not notified of your question). – Jonathan Allan May 17 '16 at 3:10
• In both 10 and 16 the numbers start with the same digit... Also 10 is one digit longer. (last code box) – ev3commander May 22 '16 at 1:04