### Re: Count up with recursive prime factorization

Posted:

**Sun Nov 30, 2014 1:37 am UTC**(.)(((..)))(.)

531

Forums for the webcomic xkcd.com

http://forums3.xkcd.com/

Page **14** of **18**

Posted: **Sun Nov 30, 2014 1:37 am UTC**

(.)(((..)))(.)

531

Posted: **Sun Nov 30, 2014 2:13 am UTC**

532

<><><<><>><<><><>> b-notation.

Before I go through the standard rigmarole I do with this (you know, ANALPHABETIC, ASYMMETRIC, COMPOSITE, <∴>-SMOOTH), I'd like to point out something interesting.

:<:><:.> dotted b-notation

If this were /2*3'd (so 768), then it would be "<><.><..><...>", where quotes represent string delimiters, not prime representations. An interesting pattern.

- levelset labynot

- boundary labynot

This is also laby-square (which minimizes area (49 = <:><:> =) and perimeter (28 = <><><:> = :)).

<><><<><>><<><><>> b-notation.

Before I go through the standard rigmarole I do with this (you know, ANALPHABETIC, ASYMMETRIC, COMPOSITE, <∴>-SMOOTH), I'd like to point out something interesting.

:<:><:.> dotted b-notation

If this were /2*3'd (so 768), then it would be "<><.><..><...>", where quotes represent string delimiters, not prime representations. An interesting pattern.

- levelset labynot

- boundary labynot

This is also laby-square (which minimizes area (49 = <:><:> =) and perimeter (28 = <><><:> = :)).

Posted: **Sun Nov 30, 2014 2:28 am UTC**

<^>

<O>

V

<^^>

{o}

VV

<O>

V

<^^>

{o}

VV

533

Posted: **Sun Nov 30, 2014 2:57 am UTC**

534

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

ab<aaab>

p1pp1p<p1p1p1pp1>

112221111123

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 11

Max Depth: 3

Factors: 3

Smoothness: 89

Necessary Brackets: 2

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

ab<aaab>

p1pp1p<p1p1p1pp1>

112221111123

Code: Select all

`* * *******`

* * * * *

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 11

Max Depth: 3

Factors: 3

Smoothness: 89

Necessary Brackets: 2

Posted: **Sun Nov 30, 2014 3:19 am UTC**

535

<<<>>><<><<><>><>> b-notation

c<a<aa>a> alphabetic b-notation

3321211212 - faubian numerical cypher b-notation

3331121112 - standard form faubian numerical cypher b-notation

3321123212 - look-and-said form of standard form faubian numerical cypher b-notation

333111212 - inverted look-and-said form of standard form faubian numerical cypher b-notation

333111122 - inverted look-and-said form of first faubian numerical cypher b-notation

2312111221121112 - look-and-said form of first faubian numerical cypher b-notation

labinot level-set

COMPOSITE, ANALPHABETIC, ASYMMETRIC

Depth: <<>>

Smoothness: <<><<><>><>>, so not really at all

Length: <><<>><<>>. Personally, I prefer the half-length (<<>><<>>), because it is the number of p's in p-notation.

<<<>>><<><<><>><>> b-notation

c<a<aa>a> alphabetic b-notation

3321211212 - faubian numerical cypher b-notation

3331121112 - standard form faubian numerical cypher b-notation

3321123212 - look-and-said form of standard form faubian numerical cypher b-notation

333111212 - inverted look-and-said form of standard form faubian numerical cypher b-notation

333111122 - inverted look-and-said form of first faubian numerical cypher b-notation

2312111221121112 - look-and-said form of first faubian numerical cypher b-notation

labinot level-set

COMPOSITE, ANALPHABETIC, ASYMMETRIC

Depth: <<>>

Smoothness: <<><<><>><>>, so not really at all

Length: <><<>><<>>. Personally, I prefer the half-length (<<>><<>>), because it is the number of p's in p-notation.

Posted: **Sun Nov 30, 2014 3:21 am UTC**

O

O((OOO))O

O((OOO))O

536

Post used to say

Posted: **Sun Nov 30, 2014 3:34 am UTC**

537

<<>><<<<<>><>>>> - b-notation

b<<<ba>>> - alphabetic b-notation

225214 - faubian cypher b-notation

- level-set labinot

- better level-set labinot

- colored level-set labinot

- larger colored level-set labinot

- hypnotic larger colored level-set labinot

COMPOSITE, AN-ALPHABETIC, ASYMMETRIC

<<<>>>-deep

<<<<<>><>>>>-smooth

<><><> p's (::-length)

<<>><<<<<>><>>>> - b-notation

b<<<ba>>> - alphabetic b-notation

225214 - faubian cypher b-notation

- level-set labinot

- better level-set labinot

- colored level-set labinot

- larger colored level-set labinot

- hypnotic larger colored level-set labinot

COMPOSITE, AN-ALPHABETIC, ASYMMETRIC

<<<>>>-deep

<<<<<>><>>>>-smooth

<><><> p's (::-length)

Posted: **Mon Dec 01, 2014 2:56 am UTC**

<><<<>><<><><>>> 538

efficiency: 66.93%

efficiency: 66.93%

Posted: **Mon Dec 01, 2014 4:11 am UTC**

(..)d(..)

539

Posted: **Mon Dec 01, 2014 4:58 am UTC**

540

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

aabbbc

p1p1pp1pp1pp1ppp1

111122222233

Prime: No

Symmetrical: No

Alphabetical: Yes

Length: 22

Reversals: 11

Max Depth: 3

Factors: 6

Smoothness: 5

Necessary Brackets: 0

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

aabbbc

p1p1pp1pp1pp1ppp1

111122222233

Code: Select all

`* * * * * *`

* * * *

*

Prime: No

Symmetrical: No

Alphabetical: Yes

Length: 22

Reversals: 11

Max Depth: 3

Factors: 6

Smoothness: 5

Necessary Brackets: 0

Posted: **Mon Dec 01, 2014 5:26 am UTC**

[[[[]]]8[[[]]]]

541

(where the center is two circles, not an 8)

Posted: **Mon Dec 01, 2014 8:07 am UTC**

faubiguy on page 7 wrote:271

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

<a<ac>>Reecer6 on page 7 wrote:I came up with 2926438, actually.

I think you have an extra '<' in the 7th position, and one missing from the 15th position. (It took quite a while to calculate that)

*2 difference

542 decimal

<><<><<><<<>>>>> standard benot

<><<<><<<>>>><>> alt. benot

a<a<ac>> alphabetic benot

a<<ac>a> alt. version

11212135 faubinumericypher benot

11313412 alt version

levelset labinot

[][[][[][[[]]]]]

COMPOSITE, ABALPHABETIC, ASYMMETRIC

<<<>>> - deep

<<><<><<<>>>>> - smooth (not smooth at all)

<<><>> - nodes (<>(:) length)

EDIT: Fixed my labinot one. I did originally, which is 202, a somewhat different number.

Posted: **Mon Dec 01, 2014 8:20 am UTC**

faubiguy wrote:181

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

<ab<aa>>

(o⊙(..))

◠

〇

⊙

<••>

◡

*3 difference.

<•>

◠

〇

⊙

<••>

◡

◠

〇

⊙

<••>

◡

543

Posted: **Mon Dec 01, 2014 8:28 am UTC**

Vytron on page 1 wrote:pp<p1p1> (17)phillip1882 wrote:p1p1p1p1

on the abiguity problem, another option is to use parenthesis.

6 = p(p1)p1, 7 = p(p1p1)

Yeah, what would work too. Though parenthesis give the illusion of multiplication (as in p*(p1p1)) while I think <>'s do a good job of showing recursion.

*32 difference = 544 - decimal

p1p1p1p1p1pp<p1p1> - penot

<><><><><><<<><>>> - benot

aaaaa<<aa>> - alph. benot

11111111113113 - faubinumericypher benot

(10)1132113 - look-and-say of faubinumericypher benot

- level set labinot

•

::

◠◠

<••>

◡◡

::

◠◠

<••>

◡◡

ABALPHABETIC, ASYMMETRIC, COMPOSITE

<<>>-deep

<<<><>>>-smooth (quite smooth for a large number!)

<<>><<>> nodes (<<>><><<>> length)

EDIT: 32 is 2^5, not 2^6. Whoops. Original labyrinthine notations: (all the others are in the same format)

Posted: **Mon Dec 01, 2014 8:47 am UTC**

Vytron wrote:Λ

O

V

ΛΛ

O

ΛΛ

O

VV

VV

327

Difference of /3 *5

ΛΛ

O

VV

ΛΛ

O

ΛΛ

O

VV

VV

O

VV

ΛΛ

O

ΛΛ

O

VV

VV

545

Posted: **Mon Dec 01, 2014 2:22 pm UTC**

<><<>><<><>><<><<>>> 546

efficiency: 63.47

efficiency: 63.47

Posted: **Mon Dec 01, 2014 11:21 pm UTC**

o

Λ

◇

⊙

VVV

547

Posted: **Tue Dec 02, 2014 1:05 am UTC**

548

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

aa<bd>

p1p1p<pp1pppp1>

11113245

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 7

Max Depth: 5

Factors: 3

Smoothness: 137

Necessary Brackets: 2

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

aa<bd>

p1p1p<pp1pppp1>

11113245

Code: Select all

`* * ***`

* *

* *

*

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 7

Max Depth: 5

Factors: 3

Smoothness: 137

Necessary Brackets: 2

Posted: **Tue Dec 02, 2014 1:19 am UTC**

⊙

<^>

⊙

{o}

v

⊙

<^>

⊙

{o}

v

⊙

549

Posted: **Tue Dec 09, 2014 1:20 am UTC**

550 - decimal 11, 2, 2, 5, 5

<><<<>>><<<<>>>><<<>>><> - benot

acdca - alph. benot

1133443311 - faubinumericypher benot

ALPHABETIC, SYMMETRIC, COMPOSITE

<><>-deep

<<<<>>>>-smooth (quite smooth for a large number!)

<><<>><> nodes (<><><><<>> length)

<><<<>>><<<<>>>><<<>>><> - benot

acdca - alph. benot

1133443311 - faubinumericypher benot

ALPHABETIC, SYMMETRIC, COMPOSITE

O

(⊙)

«⊙»

(⊙)

O

(⊙)

«⊙»

(⊙)

O

<><>-deep

<<<<>>>>-smooth (quite smooth for a large number!)

<><<>><> nodes (<><><><<>> length)

Posted: **Tue Dec 09, 2014 2:09 am UTC**

551

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

<aaa><ac>

p<p1p1p1>p<p1ppp1>

2111122134

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 9

Max Depth: 4

Factors: 2

Smoothness: 29

Necessary Brackets: 4

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

<aaa><ac>

p<p1p1p1>p<p1ppp1>

2111122134

Code: Select all

`***** ***`

* * * * *

*

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 9

Max Depth: 4

Factors: 2

Smoothness: 29

Necessary Brackets: 4

Posted: **Sat Dec 20, 2014 9:45 pm UTC**

552 = <><><><<>><<<>><<>>>

efficiency: 63.53%

efficiency: 63.53%

Posted: **Sat Dec 20, 2014 10:05 pm UTC**

Yay! Finally get to make an odd number!

553

d

◠

〇

(((〇)))

◡

◠

〇

(((〇)))

◡

553

Posted: **Sun Dec 21, 2014 1:28 pm UTC**

554

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

a<<<<aa>>>>

p1p<p<p<p<p1p1>>>>

115115

Prime: No

Symmetrical: No

Alphabetical: No

Length: 14

Reversals: 5

Max Depth: 5

Factors: 2

Smoothness: 277

Necessary Brackets: 8

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

a<<<<aa>>>>

p1p<p<p<p<p1p1>>>>

115115

Code: Select all

`* ***`

***

***

***

* *

Prime: No

Symmetrical: No

Alphabetical: No

Length: 14

Reversals: 5

Max Depth: 5

Factors: 2

Smoothness: 277

Necessary Brackets: 8

Posted: **Mon Dec 22, 2014 3:56 am UTC**

۞

(())

Λ

۞

V

(())

Λ

۞

V

555

Posted: **Mon Dec 22, 2014 4:15 am UTC**

556

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

aa<a<<aa>>>

p1p1p<p1p<p<p1p1>>>

1111213114

Prime: No

Symmetrical: No

Alphabetical: No

Length: 16

Reversals: 9

Max Depth: 4

Factors: 3

Smoothness: 139

Necessary Brackets: 6

Fixed. I forgot to add 1 to the number in the previous post.

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

aa<a<<aa>>>

p1p1p<p1p<p<p1p1>>>

1111213114

Code: Select all

`* * *****`

* ***

***

* *

Prime: No

Symmetrical: No

Alphabetical: No

Length: 16

Reversals: 9

Max Depth: 4

Factors: 3

Smoothness: 139

Necessary Brackets: 6

Vytron wrote:Did you do the same number that I did? D:

Fixed. I forgot to add 1 to the number in the previous post.

Posted: **Mon Dec 22, 2014 7:43 am UTC**

You skipped 555

Times five ((())).

So

⊙<∞۞>(⊙)

Is 555

Eth prime.

So

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

Is 557

Vytron wrote:⊙

^

∞

۞

v

111

Post used to saySpoiler:

Times five ((())).

So

⊙<∞۞>(⊙)

Is 555

phillip1882 wrote:<><<>><<<><>>> 102

Eth prime.

So

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

Is 557

Posted: **Mon Dec 22, 2014 11:36 am UTC**

558

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

abbe

p1pp1pp1ppppp1

11222255

Prime: No

Symmetrical: No

Alphabetical: Yes

Length: 20

Reversals: 7

Max Depth: 5

Factors: 4

Smoothness: 31

Necessary Brackets: 0

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

abbe

p1pp1pp1ppppp1

11222255

Code: Select all

`* * * *`

* * *

*

*

*

Prime: No

Symmetrical: No

Alphabetical: Yes

Length: 20

Reversals: 7

Max Depth: 5

Factors: 4

Smoothness: 31

Necessary Brackets: 0

Posted: **Mon Dec 22, 2014 6:31 pm UTC**

Elmach wrote:You skipped 555

What? No! You just checked my post too early

<^>

( 8 )

V

<^>

<O>

V

( 8 )

V

<^>

<O>

V

559

Posted: **Mon Dec 22, 2014 7:26 pm UTC**

560

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

aaaac<aa>

p1p1p1p1ppp1p<p1p1>

11111111332112

Prime: No

Symmetrical: No

Alphabetical: No

Length: 20

Reversals: 13

Max Depth: 3

Factors: 6

Smoothness: 7

Necessary Brackets: 2

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

aaaac<aa>

p1p1p1p1ppp1p<p1p1>

11111111332112

Code: Select all

`* * * * * ***`

* * *

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 20

Reversals: 13

Max Depth: 3

Factors: 6

Smoothness: 7

Necessary Brackets: 2

Posted: **Mon Dec 22, 2014 7:32 pm UTC**

ΛΛΛ

o

VVV

[[◇◇]]

۞

o

VVV

[[◇◇]]

۞

561

Posted: **Mon Dec 22, 2014 7:42 pm UTC**

562

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

a<aabc>

p1p<p1p1pp1ppp1>

1121112234

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 9

Max Depth: 4

Factors: 2

Smoothness: 281

Necessary Brackets: 2

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

a<aabc>

p1p<p1p1pp1ppp1>

1121112234

Code: Select all

`* *******`

* * * *

* *

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 9

Max Depth: 4

Factors: 2

Smoothness: 281

Necessary Brackets: 2

Posted: **Mon Dec 22, 2014 8:05 pm UTC**

<(⊙⊙⊙)>

563

Posted: **Mon Dec 22, 2014 8:12 pm UTC**

564

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

aab<bc>

p1p1pp1p<pp1ppp1>

1111223234

Prime: No

Symmetrical: No

Alphabetical: No

Length: 20

Reversals: 9

Max Depth: 4

Factors: 4

Smoothness: 47

Necessary Brackets: 2

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

aab<bc>

p1p1pp1p<pp1ppp1>

1111223234

Code: Select all

`* * * ***`

* * *

* *

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 20

Reversals: 9

Max Depth: 4

Factors: 4

Smoothness: 47

Necessary Brackets: 2

Posted: **Mon Dec 22, 2014 8:54 pm UTC**

(())

◠

o

⊙

◠

۞

◡

◡

◠

o

⊙

◠

۞

◡

◡

565

Posted: **Mon Dec 22, 2014 9:37 pm UTC**

566

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

a<<abb>>

p1p<p<p1pp1pp1>>

11312224

Prime: No

Symmetrical: No

Alphabetical: No

Length: 16

Reversals: 7

Max Depth: 4

Factors: 2

Smoothness: 283

Necessary Brackets: 4

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

a<<abb>>

p1p<p<p1pp1pp1>>

11312224

Code: Select all

`* *****`

*****

* * *

* *

Prime: No

Symmetrical: No

Alphabetical: No

Length: 16

Reversals: 7

Max Depth: 4

Factors: 2

Smoothness: 283

Necessary Brackets: 4

Posted: **Mon Dec 22, 2014 9:58 pm UTC**

⊙⊙

(..)

⊙⊙

(..)

⊙⊙

567

Posted: **Tue Dec 23, 2014 12:40 am UTC**

568

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

aaa<aac>

p1p1p1p<p1p1ppp1>

111111211134

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 11

Max Depth: 4

Factors: 4

Smoothness: 71

Necessary Brackets: 2

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

aaa<aac>

p1p1p1p<p1p1ppp1>

111111211134

Code: Select all

`* * * *****`

* * *

*

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 18

Reversals: 11

Max Depth: 4

Factors: 4

Smoothness: 71

Necessary Brackets: 2

Posted: **Tue Dec 23, 2014 1:02 am UTC**

Λ

Λ

۞

V

O

V

Λ

۞

V

O

V

569

Posted: **Tue Dec 23, 2014 1:49 am UTC**

570

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

abc<aaa>

p1pp1ppp1p<p1p1p1>

112233211112

Prime: No

Symmetrical: No

Alphabetical: No

Length: 20

Reversals: 11

Max Depth: 3

Factors: 4

Smoothness: 19

Necessary Brackets: 2

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

abc<aaa>

p1pp1ppp1p<p1p1p1>

112233211112

Code: Select all

`* * * *****`

* * * * *

*

Prime: No

Symmetrical: No

Alphabetical: No

Length: 20

Reversals: 11

Max Depth: 3

Factors: 4

Smoothness: 19

Necessary Brackets: 2