## Count up with recursive prime factorization

For all your silly time-killing forum games.

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Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

AarexTiaokhiao (page 7) wrote:217
<<><>><<<<>>>>
(oo)((⊙))

Code: Select all

`  |  ||||| |`

*3 makes
651.

651 = <<>> <<><>> <<<<>>>>
= b(aa)d

The quoted post used an old notation. In that notation (vertical lineospacial?), 651 is represented by

Code: Select all

`   |   ||||||| |`

. In the depth notation (which came about later) we have

Code: Select all

`* *** ** * * *      *      *`

As can be clearly seen from the depth notation, the max depth is four, it has three prime factors, and the representation can neither be symmetric nor alphabetic.

Only one pair of brackets is needed; this can also be seen from the depth notation.

Posts: 49
Joined: Thu Jun 25, 2015 10:43 am UTC

### Re: Count up with recursive prime factorization

faubiguy, on page 4 wrote:163
<<><<><><>>>
<o<ooo>>

〇〇〇

*4 makes 652.

<><><<><<><><>>>
This is a signature, in case you didn't notice.

faubiguy
Posts: 15
Joined: Sun Aug 11, 2013 9:20 am UTC

### Re: Count up with recursive prime factorization

653
<<<><>><<<><>>>>
<<aa><<aa>>>
p<p<p1p1>p<p<p1p1>>>
31123114

Code: Select all

`********** **** * ***    * *`
$\mathfrak{p}(\omega^{\omega^{\omega^{2}}+\omega^{2}})$

Prime: Yes
Symmetrical: No
Alphabetical: No

Length: 16
Reversals: 7
Max Depth: 4
Factors: 1
Smoothness: 653
Necessary Brackets: 8

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

Vytron on page 8 in 2014 wrote:
Λ
O
V
ΛΛ
O
ΛΛ
O
VV
VV

327

*2 is 654. For those who don't recognize this type of notation, although it appears to have a two-dimensional structure, you read it from left to right, top to bottom, so this should read as
OPEN (or <), OH (or <>), CLOSE (or >), OPEN OPEN, OH, OPEN OPEN, OH, CLOSE CLOSE, CLOSE CLOSE,
which is <<>><<<><<<>>>>>, which is, of course, correct.

654
<><<>><<<><<<>>>>>
ab<<ac>>
pdd<pf>
11223135

Code: Select all

`* * ***  * ***    * *      *      *`

$\mathfrak{p}(\omega^{\omega^{\omega^{\omega}+1}}+\omega+1)$

654 in level-set notation; if you run this through an edge-detector, you will get it in boundary notation
654l.png (1.22 KiB) Viewed 4067 times

Prime: No
Prime Power: No
Symmetrical: No
Alphabetical: No
Square-free: Yes
Recursively Square-free: Yes (all terms are composed only of omegas and ones, in (condensed) ordinal notation)

Nodes: 9
Reversals: 7
Max Depth: 5
Factors: 3
Smoothness: 109
Necessary Brackets: 4

faubiguy
Posts: 15
Joined: Sun Aug 11, 2013 9:20 am UTC

### Re: Count up with recursive prime factorization

655
<<<>>><<><><><><>>
c<aaaaa>
ppp1p<p1p1p1p1p1>
332111111112

Code: Select all

`* ********** * * * * **          `
$\mathfrak{p}(\omega^{5}+\omega^{\omega})$

Prime: No
Symmetrical: No
Alphabetical: No

Length: 18
Reversals: 11
Max Depth: 3
Factors: 2
Smoothness: 131
Necessary Brackets: 2

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

edit: ninjaed
656
<><><><><<<><<>>>>
66.1 % efficient
good luck have fun

faubiguy
Posts: 15
Joined: Sun Aug 11, 2013 9:20 am UTC

### Re: Count up with recursive prime factorization

657
<<>><<>><<<>><<><>>>
bb<b<aa>>
pp1pp1p<pp1p<p1p1>>
2222322113

Code: Select all

`* * ****** * * ***    * * *`

$\mathfrak{p}(\omega^{\omega^{2}+\omega}+\omega2)$

Prime: No
Prime Power: No
Symmetrical: No
Alphabetical: No
Square-free: No
Recursively Square-free: No

Length: 20
Reversals: 9
Max Depth: 3
Factors: 3
Smoothness: 73
Necessary Brackets: 4

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

658

2<1>7<2<1>2<1>>47<3<2<1>>>5<3<2<1>>>>>
<><<><>><<<>><<<>>>>
<><aa><bc>

some cool information

ωωω2+1

faubiguy
Posts: 15
Joined: Sun Aug 11, 2013 9:20 am UTC

### Re: Count up with recursive prime factorization

659
<<><><><<>><<<>>>>
<aaabc>
p<p1p1p1pp1ppp1>
2111112234

Code: Select all

`********** * * * *      * *        *`

$\mathfrak{p}(\omega^{\omega^{\omega}+\omega+3})$

Prime: Yes
Prime Power: Yes
Symmetrical: No
Alphabetical: No
Square-free: Yes
Recursively Square-free: No

Length: 18
Nodes: 18
Reversals: 9
Max Depth: 4
Factors: 1
Smoothness: 659
Necessary Brackets: 2

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

faubiguy wrote:655
$\mathfrak{p}(\omega^{5}+\omega^{\omega})$

There's a bug in your code; you want ωω + ω5 because ordinal arithmetic.

660 = 2 3 3 5 11 = abbcd

<><<>><<>><<<>>><<<<>>>>

ω⬆️⬆️4 + ω⬆️⬆️3 + ω⬆️⬆️2 2 + ω⬆️⬆️1

Code: Select all

`* * * * ** * * ** **`

Composite, alphabetic, asymmetric, cube-free

faubiguy
Posts: 15
Joined: Sun Aug 11, 2013 9:20 am UTC

### Re: Count up with recursive prime factorization

661
<<<<<>>>><<<<>>>>>
<dd>
p<pppp1pppp1>
5445

Code: Select all

`**** ** ** ** *`

Prime: Yes
Prime Power: Yes
Symmetrical: Yes
Alphabetical: No
Square-free: Yes
Recursively Square-free: No

Length: 18
Nodes: 18
Reversals: 3
Max Depth: 5
Factors: 1
Smoothness: 661
Necessary Brackets: 2

Elmach wrote:There's a bug in your code; you want ωω + ω5 because ordinal arithmetic.

You're right. I've taken that part out for now until I decide to figure out how to correct that. The rest of the calculations all use commutative operations so I think I just forgot to take the ordering into account.

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

662
<><<<<><><>>>>
a<<<aaa>>>
pf<ppp>
11411114

Code: Select all

`* *****  *****  *****  * * *`

$\mathfrak{p}(\omega^{\omega^{\omega^{3}}}+1)$

Prime: No
Prime Power: No
Symmetrical: No
Alphabetical: No

Nodes: 7
Reversals: 7
Max Depth: 4
Factors: 2
Smoothness: 67 331 EDIT: I found a bug in my program, and only noticed this was wrong on Mar 13
Necessary Brackets: 6
----
faubiguy wrote:
Elmach wrote:There's a bug in your code; you want ωω + ω5 because ordinal arithmetic.

You're right. I've taken that part out for now until I decide to figure out how to correct that. The rest of the calculations all use commutative operations so I think I just forgot to take the ordering into account.

I actually fixed that bug in my code when I rewrote yours to make use of classes (because I like OOP).

I have no idea how my fix can be implemented in your code, because my fix relies completely on using classes.

EDIT: The bug is that your prime factorization would factor 105 as 3*5*7 = <<>><<<>>><<><>> = bc<aa>.
Doing the obvious algorithm from there would give w2 + ww + w.
My code would factor 105 as 3*7*5 (<<>><<><>><<<>>> = b<aa>c), which gives ww + w2 + w.
Last edited by Elmach on Mon Mar 13, 2017 9:56 pm UTC, edited 1 time in total.

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

663
<<>><<><<>>><<<><>>>
good luck have fun

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

664
<><><><<<<>><<>>>>
aaa<<bb>>1
pppd(dd)2
1111114224

Code: Select all

`* * * ***      ***      * *      * *`

$\mathfrak{p}(\omega^{\omega^{\omega2}}+3)$
terprimus et secundus bissecundi
Labyrinth notation, level-set mode. Run through an edge detector to get it in boundary mode.
664l.png (481 Bytes) Viewed 4007 times
11 x 13 // 9 x 11 size - closest to square (minimum square is 13 x 13 // 11 x 11)
664lalt.png (405 Bytes) Viewed 4006 times
9 x 15 // 7 x 13 - smallest area, same perimeter
Quarter-Perimeter: 6 = <><<>> = ab1 // 5 = <<<>>> c1

Composite
Asymmetric
Not Alphabetic

Nodes: 9 = <<>><<>> = bb1
Reversals: 9 = <<>><<>> = bb1
Max Depth: 4 = <><> = aa1
Factors: 2 = <> = a1Umm, it's obviously 4 = <><> = aa??? How did I miss this bug? (From Mar 13)
Smoothness: 2383 = <<<<>><<>>>> = d(dd)2 oddly enough, this part of the number is correct, because I did it manually.
Necessary Brackets: 4 = <><> = aa1

----
1 bracket notation using alphabetic extension - replace <> with a, <<>> with b, etc.
2 prime notation using spdf extension - use parentheses, replace pp with d, ppp with f, etc; 1 -> s, p1 -> p, d1 -> d, etc.
Last edited by Elmach on Mon Mar 13, 2017 9:58 pm UTC, edited 2 times in total.

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

665
<<<>>><<><>><<><><>>
c<aa><aaa>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

Some people might call this number...

a lot of sixes.

666
<><<>><<>><<><><<>>>
abb<aab>
pddP(ppd)
112222211123

Code: Select all

`* * * *****  * * * * *          *`

$\mathfrak{p}(\omega^{\omega+2}+\omega2+1)$
primus et bissecundus et primus (bisprimi et secundi)
I don't think this worked that well. 11 by 11 (9 by 9).
666l.png (1.14 KiB) Viewed 3917 times
This one is 13 by 9 (11 by 7)
666lalt1.png (1.1 KiB) Viewed 3914 times

Heavily Stylized
666lalt2.png (1.84 KiB) Viewed 3914 times

Composite
Asymmetric
Not Alphabetic
3-free, even recursively!

Nodes: 10
Reversals: 11
Max Depth: 3
Factors: 4
Smoothness: ... 37, the 12th prime. I see a bug in my code now - I will go fix it shortly.
Necessary Brackets: 2
-----------------------------------------
redundakitty
Spoiler:
666alt1.png (190 Bytes) Viewed 3914 times

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

667
<<<>><<>>><<><<<>>>>
<bb><ac>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

668
<><><<<>><<><<>>>>
aa<b<ab>>
ppP(dP(pd))
1111322124

Code: Select all

`* * *****    * ***    * * *        *`

$\mathfrak{p}(\omega^{\omega^{\omega+1}+\omega}+2)$
bisprīmus et prīmus (secundī et prīmī (prīmī et secundī))
15 by 9 (13 by 7) - can fit one more dot for 1336
668ld.png (6.56 KiB) Viewed 3907 times

Composite
Asymmetric
Not Alphabetic
3-free, even recursively!

Nodes: 9
Reversals: 9
Max Depth: 4
Factors: 3
Smoothness: 167
Necessary Brackets: 4

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

669
<<>><<><><><><<>>>
66.23% efficiency
good luck have fun

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

670
<><<<>>><<<><><>>>
ac<<aaa>>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

671
<<<<>>>><<><<>><<>>>
64.62% efficiency
good luck have fun

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

672
<><><><><><<>><<><>>
aaaaab<aa>
pppppdP(pp)
1111111111222112

Code: Select all

`* * * * * * ***          * * *`

$\mathfrak{p}(\omega^{2}+\omega+5)$
(hey! a bug! fixed.(?)
(Edit:.. this is a magic bug - it only occurs half the time, randomly...)
quīnquiēnsprīmus et prīmus bisprīmī et secundus
11 by 7 (9 by 5) perfectly
672ld.png (5.21 KiB) Viewed 3872 times

Composite
Asymmetric
Not Alphabetic

Nodes: 10
Reversals: 15
Max Depth: 2
Factors: 7
Smoothness: 7
Necessary Brackets: 2
----
Spoilered for wrong number (/3*5'd.)
Spoiler:
11 by 9 (9 by 7) perfectly EDIT: WRONG NUMBER (1120). FIXING
672ld.png (6.55 KiB) Viewed 3877 times

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

673
<<><<><<>><<>>>>
<a<abb>>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

674
<><<><><<<><>>>>
it occurs to me i'm measuring efficiency wrong. i should be measuring it based on e, rather than 10.
81.42% efficiency.
good luck have fun

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

phillip1882 wrote:674
<><<><><<<><>>>>
it occurs to me i'm measuring efficiency wrong. i should be measuring it based on e, rather than 10.
81.42% efficiency.

Unless you changed your definition of efficiency, I do not believe your definition required a specific base.
(Old definition was 1 - log(half-length)/log(n))

675 is 25 x 27, so

<<>><<<>>><<>><<<>>><<>>
bcbcb
2233223322
2222223333
ωω2+ ω3
tersecundus et bistertius
13 by 11 exactly (11 by 9)
675l.png (3.02 KiB) Viewed 3843 times
In retrospect, it makes sense why this version is larger in size; there are less blocks of one color. Still, two and a half times??
675ld.png (8.86 KiB) Viewed 3842 times

Obviously composite, symmetric, and alphabetic.
Nodes(Half-length): 12

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

yes i changed my definition as well.
new definition ln(n)/(len(r(n))/2)
since the recursive numbers are closely related to the primes this gives a much more accurate view of efficiency.
676
<><><<><<>>><<><<>>>
efficiency 65.16%
good luck have fun

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

677
<<<>><<<><<>>>>>
<b<<ab>>>
P(dD(pd))
323125

Code: Select all

`****** **** ***  * *    *`

$\mathfrak{p}(\omega^{\omega^{\omega^{\omega+1}}+\omega})$
prīmus (secundī et secundī (prīmī et secundī))
15 by 13 (13 by 11), with 7 dots left over.
677l.png (3.25 KiB) Viewed 3827 times

Prime
Asymmetric
Not Alphabetic
Squarefree, even recursively!

Nodes: 8
Reversals: 5
Max Depth: 5
Factors: 1 (prime)
Smoothness: 677 (still prime)
-----
I was about to say that alphabetic numbers are the same as recursively 1-free, but that's wrong - it's the same as recursively prime. (Also, the only 1-free number is 1.)

Obviously, labinot notation1, level set mode can be shown with these dots specified black and white (and the smallest area required is always of this form):
extend as needed
grid.png (10.79 KiB) Viewed 3824 times

Thus, only the dots in between need to be specified. I have not yet figured out how to do that with a program.
this version uses the 7 remaining dots to form arms
677armed.png (3.51 KiB) Viewed 3824 times

1(labyrinth notation notation, yes...)

faubiguy
Posts: 15
Joined: Sun Aug 11, 2013 9:20 am UTC

### Re: Count up with recursive prime factorization

678
<><<>><<><<>><<<>>>>
ab<abc>
p1pp1p<p1pp1ppp1>
1122212234

Code: Select all

`* * *****  * * * *      * *        *`

Prime: No
Prime Power: No
Symmetrical: No
Alphabetical: No
Square-free: Yes
Recursively Square-free: Yes

Length: 20
Nodes: 20
Reversals: 9
Max Depth: 4
Factors: 3
Smoothness: 113
Necessary Brackets: 2

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

679
<<><>><<<<>>><<<>>>>
<aa><cc>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

680
<><><><<<>>><<<><>>>
aaac<<aa>>
pppfD(pp)
111111333113
Mystery: 94677

Code: Select all

`* * * * ***      * ***      * * *`

$\mathfrak{p}(\omega^{\omega^{2}}+\omega^{\omega}+3)$
terprīmus et tertius et secundus bisprīmī
2 * (6 by 4) ± 1 size -- minimum area and perimeter
680l6.png (2.66 KiB) Viewed 3812 times

Not sure which version looks better
680l2.png (2.66 KiB) Viewed 3812 times

Composite
Asymmetric
Not Alphabetic

Nodes: 10 (half-length = number of nodes in the tree = number of different connected areas in labinot (excluding the boundary))
Reversals/extrema: 11
Max Depth: 3
Factors: 5
Smoothness: 17
Necessary Brackets: 4
-----------------------------------------

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

681
<<>><<<><>><<><>>>
b<<aa><aa>>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

682
<><<<<>>>><<<<<>>>>>
65.25% efficiency
good luck have fun

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

683
<<><<><<>><<>>>>
<a<abb>>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

684
<><><<>><<>><<><><>>
aabb<aaa>
ppddP(ppp)
11112222211112
Mystery: 176949

Code: Select all

`* * * * *****    * * * * *`

$\mathfrak{p}(\omega^{3}+\omega2+2)$
bisprīmus et bissecundus et prīmus terprīmī
5 by 4¹ - minimum perimeter of 18 (Area is 20). Two dots unused, meaning actual area is 18.
684l.png (2.37 KiB) Viewed 3797 times
This version minimizes area with a size of 9 by 21, or 18, but has a perimeter of 22.
Composite
Symmetric
Not Alphabetic

Nodes: 10
Reversals: 13
Max Depth: 2
Necessary Brackets: 2
Redundant Kittens:
Spoiler:

684al.png (905 Bytes) Viewed 3797 times

----
1 I will now be noting the size of the labyrinth notation in "dots", which are about twice the size that I was measuring it in previously. Thus, instead of saying 11 by 9 (9 by 7), I will say 5 by 4.
Last edited by Elmach on Sun Mar 26, 2017 1:38 am UTC, edited 1 time in total.

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

685
<<<>>><<<>><<<<>>>>>
c<bd>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
Contact:

### Re: Count up with recursive prime factorization

686
<><<><>><<><>><<><>>
65.31% efficiency
good luck have fun

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

687
<<>><<><<<>>><<<>>>>
b<acc>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

686
<><<><>><<><>><<><>>
a<aa><aa><aa>
pP(pp)P(pp)P(pp)
11211221122112
Mystery: 183085

Code: Select all

`* *** *** ***  * * * * * *`

$\mathfrak{p}(\omega^{2}3+1)$
prīmus et terprīmus bisprīmī
5 by 4 with 1 extra dot - however, 19 is prime, and does not have a pair of factors greater than or equal to 2, the Max Depth. Thus, this is the minimum area and perimeter possible.
Composite
Asymmetric
Not Alphabetic

Nodes: 10
Reversals: 13
Max Depth: 2
Smoothness: 7
Necessary Brackets: 6
686l.png (1.03 KiB) Viewed 3792 times

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Count up with recursive prime factorization

Ninja'd
688
<><><><><<><<><>>>
aaaa<a<aa>>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Elmach
Posts: 155
Joined: Sun Mar 13, 2011 7:47 am UTC

### Re: Count up with recursive prime factorization

Whoops, didn't notice.
689 and 690, to re-synchronize.
689 wrote:<<><><><>><<><<>>>
<aaaa><ab>
P(pppp)P(pd)
211111122123
Mystery: 27819

Code: Select all

`******* **** * * * * *          *`

$\mathfrak{p}(\omega^{\omega+1}+\omega^{4})$
prīmus quaterprīmī et prīmus (prīmī et secundī)
7 by 3 with one dot extra. I don't think this number can be represented in a 5 by 4 grid, so this has the minimum area and perimeter.
689l.png (1.06 KiB) Viewed 3767 times

Semiprime
Asymmetric
Not Alphabetic

Nodes: 9
Reversals: 11
Max Depth: 3
Factors: 2
Smoothness: 53
Necessary Brackets: 4

690 wrote:<><<>><<<>>><<<>><<>>>
abc<bb>
pdfP(dd)
1122333223
Mystery: 422349

Code: Select all

`* * * ***  * * * *    * * *`

$\mathfrak{p}(\omega^{\omega2}+\omega^{\omega}+\omega+1)$
prīmus et secundus et tertius et prīmus bissecundī
6 by 5 with one dot extra. 29 is prime, so this has the minimum area and perimeter.
690l.png (1.48 KiB) Viewed 3767 times

Composite
Asymmetric
Not Alphabetic
square-free, but recursively only cube-free

Nodes: 11
Reversals: 9
Max Depth: 3
Factors: 4
Smoothness: 23
Necessary Brackets: 2