## Count up with recursive prime factorization

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Vytron
Posts: 429
Joined: Mon Oct 19, 2009 10:11 am UTC
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### Re: Count up with recursive prime factorization

___
◜   _   ◝
¦ ◜ ◇ ◝ ¦
¦ ¦ ◇ ¦ ¦
¦ ¦ ◠ ¦ ¦
¦ ¦ ◇ ¦ ¦
¦ ¦ ◇ ¦ ¦
¦ ◟◞ ¦
◟_____◞

571

phillip1882
Posts: 145
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### Re: Count up with recursive prime factorization

572 = <><><<<<>>>><<><<>>>
efficiency:60.86%
good luck have fun

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

### Re: Count up with recursive prime factorization

573
<<>><<<><<><>>>>
b<<a<aa>>>
pp1p<p<p1p<p1p1>>>
22312114

Code: Select all

`* ****** *****  * ***    * *`

Prime: No
Symmetrical: No
Alphabetical: No

Length: 16
Reversals: 7
Max Depth: 4
Factors: 2
Smoothness: 191
Necessary Brackets: 6

phillip1882
Posts: 145
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Location: geogia
Contact:

### Re: Count up with recursive prime factorization

574 = <><<><>><<<><<>>>>
efficiency: 65.41%
good luck have fun

Vytron
Posts: 429
Joined: Mon Oct 19, 2009 10:11 am UTC
Location: The Outside. I use She/He/Her/His/Him as gender neutral pronouns :P

### Re: Count up with recursive prime factorization

<<◇>>
<۞۞>
<<◇>>

575

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

### Re: Count up with recursive prime factorization

576
<><><><><><><<>><<>>
aaaaaabb
p1p1p1p1p1p1pp1pp1
1111111111112222

Code: Select all

`* * * * * * * *            * *`

Prime: No
Symmetrical: Yes
Alphabetical: Yes

Length: 20
Reversals: 15
Max Depth: 2
Factors: 8
Smoothness: 3
Necessary Brackets: 0

Vytron
Posts: 429
Joined: Mon Oct 19, 2009 10:11 am UTC
Location: The Outside. I use She/He/Her/His/Him as gender neutral pronouns :P

### Re: Count up with recursive prime factorization

"
o
<oooo>
"

577

Where I am using quotes as another pair of angle brackets.

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

### Re: Count up with recursive prime factorization

578
<><<<><>>><<<><>>>
a<<aa>><<aa>>
p1p<p<p1p1>>p<p<p1p1>>
1131133113

Code: Select all

`* *** ***  *** ***  * * * *`

Prime: No
Symmetrical: Yes
Alphabetical: No

Length: 18
Reversals: 9
Max Depth: 3
Factors: 3
Smoothness: 17
Necessary Brackets: 8

I thought that I was the last poster on this thread, else I would have posted in the last 2 months...

phillip1882
Posts: 145
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### Re: Count up with recursive prime factorization

<<>><<><><<<<>>>>> 579
efficiency: 65.46%
good luck have fun

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

### Re: Count up with recursive prime factorization

Spoiler:
Elmach (page 2) wrote:56 (decimal)
o<..><>o (Vytronian circle/dot bracket-notation)

x ac (10)

is

00<8><☉>00 (560)

Analphabetical, asymmetric, composite
<8>, or <<><>>-smooth
<><<>> factors

EDIT: Whoops.
Vytron on page 2 wrote:
Elmach wrote:EDIT POST-NINJA: I don't get it?

<> = ◇
<<>> = ⊙
<<><>> = (..)
<<><><>> = (...)

o
^
o
^

v
v

58 in Vertical notation.

times 10 (ac) is
0<0C>0C

analphabetic, asymmetric, composite,
<ac>-smooth
<><> factors

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

### Re: Count up with recursive prime factorization

581
<<><>><<<<>><<>>>>
<aa><<bb>>
p<p1p1>p<p<pp1pp1>>
21124224

Code: Select all

`*** **** * ***    * *    * *`

Prime: No
Symmetrical: No
Alphabetical: No

Length: 18
Reversals: 7
Max Depth: 4
Factors: 2
Smoothness: 83
Necessary Brackets: 6

phillip1882
Posts: 145
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Location: geogia
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### Re: Count up with recursive prime factorization

<><<>><<<<>>><<<>>>> 582
good luck have fun

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

### Re: If you can count two two, you can count two anything wit

583
<<<<>>>><<><><><>>
d<aaaa>
pppp1p<p1p1p1p1>
4421111112

Code: Select all

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

Prime: No
Symmetrical: No
Alphabetical: No

Length: 18
Reversals: 9
Max Depth: 4
Factors: 2
Smoothness: 53
Necessary Brackets: 2

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

### Re: Count up with recursive prime factorization

Elmach on page 2 wrote:73
<b(..)> (faubi-abc, Vytronic-dot BN)
<⊙⚇> (Elmaxi/Vytronic-circle BN, using the mysterious symbol "⚇")

times eight makes 584

<⊙⚇>ooo

is not alphabetic, or symmetric, or prime.

322113111111 in faubicypher.

<><><><<<>><<><>>> for those of you who want real brackets.
111111322113 in faubicyph (that representation)
6113222113 once l&s'ed
162113322113 twice l&s'ed
051002000000 difference between twice l&s and faubicypher. Odd that it is so similar.
OOO <(()) (()())>
This would be so good in a labinot, but I am on mobile. Sigh.

phillip1882
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### Re: Count up with recursive prime factorization

<<>><<>><<<>>><<><<>>> 585
good luck have fun

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

### Re: Count up with recursive prime factorization

<><<><<<<<>>>>>> (586)
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

Vytron on page one wrote:Yeah.

Elmach wrote:(Also, maybe use parens or other grouping symbols... brackets are kind of hard to read.)

((())(()))?
([<>][<>])?
<[<>][<>]>?

I suggest using ◇ as a shorthand for <>, I think that'd make them easier to read.

◇◇<◇◇> (28)

-eth prime (107)-eth prime (587)

((()(()())())) (benot 587)
<<o⚇o>> (benot 587)

It is not alphabetic. However, it is symmetric and (doubly) prime.

Its half length is ⚇ (benot 7).
Its depth is : (benot 4).

Let s<n> = n, for all n.
Let p<n> be the nth prime.
Let d<n> be p<p<n>>.
Let this sequence of letters continue in the obvious way.

Then we have d<p<>p<p<>p<>>p<>>. I kind of hoped this would turn out better. Pe-notation never turns out well.

Take the (be-notation) representation
<<<<><>><><>>> (benot 587)
And do the faubian cypher.
41121113 (benot fc 587)
And then reverse look&say it.
1111213 (benot fc -l&s 587)
And put a zero in front so we can 1/l&s it again.
01111213 (???)
0123 (???)
I don't know what the point of that was.
Forward l&s gives
1421123113. (benot fc l&s 587)

Clearly, the length of the faubicypher negative once l&sed is the half length of the number.
Also clearly, the number of reversals is the length of the faubicypher minus one.

Also, we have
<<<:>:>> (benot 587)

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

### Re: Count up with recursive prime factorization

<><><<>><<><>><> (588)
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

phillip1882
Posts: 145
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Location: geogia
Contact:

### Re: Count up with recursive prime factorization

<<><><>><<<<<>>>>>589
good luck have fun

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

### Re: Count up with recursive prime factorization

<><<<>>><<<<><>>>> 590
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

Vytron on page 5 wrote:
<<<>><<<>>><<>>>

197

Times 3 makes 591
<⊙<⊙>⊙>⊙ (benot, ⊙=3)
((())((()))(()))(()) (benot)

(((()))(())(()))(()) (benot)
(cbb)b (benot alpha)
p<f<>d<>d<>>d<> (spdf penot)

Not alphabetic, not prime, not symmetric.

The half length is <<<>>><>, or ten.
The depth is <><>, or four.
It has () (benot two) factors.

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

### Re: Count up with recursive prime factorization

592 = 2^4*37 = <><><><><<><><<>>> = 88(8⊙)

Does anyone mind explaining the various notations and properties for me?
This is a signature, in case you didn't notice.

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

### Re: Count up with recursive prime factorization

-eth prime is 593, so we have

<<><<>><<>><<>><>> (593 benot),
Making this number symmetric, because it can be represented in RPF symmetrically.

Benot is my term for bracket-notation, or b-notation, or b-not., or benot.
Penot is p-notation, say p<p<>d<>d<>d<>p<>> (penot with spdf extension). This is mostly only used historically; it was the original notation used.

Various people make extensions to those two standard notations: most common are random symbols, based off of their shape, and the alphabet, with a, b, c, d, etc representing 2,3,5,11, etc.
Recently, I decided to try porting abc-ext. to penot, and made spdf-ext., described above.

Labinot (Labyrinthine notation, Laby-not., labinot) is a specific pictorial way. After some examples, it should be obvious.

Any notation that shows the structure is valid. Some people like making new notations and let them be explained by logic.

593 is prime, and thus, not composite.
It cannot be represented using only the abc extension, and thus is not alphabetic.
593 is symmetric, however.

Because 593 is prime, it is 593-smooth. Smoothness is defined as the largest prime factor. See Wikipedia.
The length is the number of brackets. I prefer the half length, which for this number, is <<>><<>>, or nine.
The depth is how deep the notation is; not sure how to explain it further. 593 is <<>>-deep.

Faubiguy, a long time ago, decided to represent the structure by looking at the number of brackets. Thus, one possible representation is
2122222212 (benot faubicypher 593)
The look and say sequence is generated by looking at each term, and counting how many of it there is. See Wikipedia. Once l&s'ed, we get
12 11 62 11 12 (benot faubicyph l&s 583)
12116211112, abbreviated, which is how Conway intended it to be.

It is not necessary to post all the information I do.
It should be noted that a lot of the novel notations become clearer the more you understand RPF. You begin to see the structure, and from there, you see how the notation arose.

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

### Re: Count up with recursive prime factorization

594 = 2*3^3*11 = <><<>><<>><<>><<<<>>>> = o⊙⊙⊙((⊙)) = abbbd

Depth 4, half-length 11, alphabetic, asymettric, 11-smooth, not prime, 5 factors
This is a signature, in case you didn't notice.

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

### Re: Count up with recursive prime factorization

595 is 5 times 7 times 17
Is trisprime times prime of sq prime times bisprime of sq prime (i could have sworn i have said something like that in this thread, but it appears not.)

(Edit from future: I would have said something like tertius times primus bisprimi times secundus bisprimi.)

Or in standard notations,
<<<>>><<><>><<<><>>> (benot, 595)
2221123113 (benot, faubicypher, 595)
(SUN) DIE-FACE-TWO (DIE-FACE-TWO) (benot, symbols, yelling, 595)
((•))(:)((:)) (benot, symbols, 595)
c<aa><<aa>> (benot, abc-ext., 595)

f<>p<p<>p<>>d<p<>p<>> (penot, spdf ext., 595)
fp(pp)d(pp) (penot, spdf with an obvious fix, 595)

Commentary on 595:
Not prime, not alphabetic, not symmetric.
<<>>-deep (3-deep)
d(pp)-smooth (17-smooth)
b factors (3 factors)
<><<<>>><>-long (10 half-length, 20 length)

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

### Re: Count up with recursive prime factorization

596
<><><<<<>>><<><>>>
aa<c<aa>>
p1p1p<ppp1p<p1p1>>
1111432113

Code: Select all

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

Prime: No
Symmetrical: No
Alphabetical: No

Length: 18
Reversals: 9
Max Depth: 4
Factors: 3
Smoothness: 149
Necessary Brackets: 4

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

### Re: Count up with recursive prime factorization

AarexTiaokhiao on page 5 wrote:199
<<><<<>><<>>>>
(o(⊙⊙))

Code: Select all

`****** ***  * *  * *`

*3 makes 597.

<<><<<>><<>>>><<>> = (o(⊙⊙))⊙
Half length 9, 2 factors, depth 4, 199-smooth
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

598
<><<><<>>><<<>><<>>>
a<ab><bb>
p1p<p1pp1>p<pp1pp1>
1121233223

Code: Select all

`* *** ***  * * * *    * * *`

Prime: Yes
Symmetrical: No
Alphabetical: No

Length: 20
Reversals: 9
Max Depth: 3
Factors: 3
Smoothness: 23
Necessary Brackets: 4

Apparently these recursive prime factorization numbers are also known as the Matula numbers of rooted trees, with the rooted tree being the bracket notation. Each pair of brackets is a node with contained pairs of brackets as its children, and the outermost pair of brackets as the root.

Edit: Actually, the root would be the imaginary parent node that contains all of the top level factors

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

### Re: Count up with recursive prime factorization

On page 3, Vytron wrote:
^
<◠>
<<◇>>

V

109[/code]

th prime = 599.

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

phillip1882
Posts: 145
Joined: Fri Jun 14, 2013 9:11 pm UTC
Location: geogia
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### Re: Count up with recursive prime factorization

<><><><<>><<<>>><<<>>>
600
good luck have fun

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

### Re: Count up with recursive prime factorization

601
<<><<<>>><<<<>>>>>
<acd>
p<p1ppp1pppp1>
213345

Code: Select all

`****** * *  * *  * *    *`

Prime: Yes
Symmetrical: No
Alphabetical: No

Length: 18
Reversals: 5
Max Depth: 5
Factors: 1
Smoothness: 601
Necessary Brackets: 2

emlightened
Posts: 42
Joined: Sat Sep 26, 2015 9:35 pm UTC
Location: Somewhere cosy.

### Re: Count up with recursive prime factorization

<><<><>><<><<><>>> (602)
14p14
$\mathfrak{p}(\omega^{\omega^{2}+1}+\omega^{2}+1)$

The last one's prime ordinal representation. Express the ordinal solely in terms of 0 and $\alpha\mapsto\omega^\alpha$, then map each 0 to a 1, each + to a * and each $\alpha\mapsto\omega^\alpha$ to $\alpha\mapsto p_\alpha$. (If you don't know what that is, it's ordinals, used in set theory. Look up ordinal arithmetic on Wikipedia for more info; it may or may not help.)

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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

### Re: Count up with recursive prime factorization

603
<<>><<>><<<><><>>>
bb<<aaa>>
pp1pp1p<p<p1p1p1>>
2222311113

Code: Select all

`* * ****** * *****    * * *`

Prime: No
Symmetrical: Yes
Alphabetical: No

Length: 18
Reversals: 9
Max Depth: 3
Factors: 3
Smoothness: 67
Necessary Brackets: 4

emlightened
Posts: 42
Joined: Sat Sep 26, 2015 9:35 pm UTC
Location: Somewhere cosy.

### Re: Count up with recursive prime factorization

604
<><<><<>><<>><>><>
a<abba>a
$\mathfrak{p}(\omega^{\omega2+2}+2)$

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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

### Re: Count up with recursive prime factorization

605
<<<>>><<<<>>>><<<<>>>>
cdd
ppp1pppp1pppp1
334444

Code: Select all

`* * ** * ** * *  * *`

Prime: No
Symmetrical: Yes
Alphabetical: Yes

Length: 22
Reversals: 5
Max Depth: 4
Factors: 3
Smoothness: 11
Necessary Brackets: 0

emlightened
Posts: 42
Joined: Sat Sep 26, 2015 9:35 pm UTC
Location: Somewhere cosy.

### Re: Count up with recursive prime factorization

606
<><<>><<><<><<>>>>
ab<a<ab>>
$\mathfrak{p}(\omega^{\omega^{\omega+1}+1}+\omega+1)$

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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

### Re: Count up with recursive prime factorization

607
<<<>><<<>><><>>>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

emlightened
Posts: 42
Joined: Sat Sep 26, 2015 9:35 pm UTC
Location: Somewhere cosy.

### Re: Count up with recursive prime factorization

608
<><><><><><<><><>>
aaaaa<aaa>
$\mathfrak{p}(\omega^3+5)$

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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

### Re: Count up with recursive prime factorization

609
<<>><<><>><<><<<>>>>
c<aa><ac>
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

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

### Re: Count up with recursive prime factorization

610
<><<<>>><<><<>><<>>>
ac<abb>
p1ppp1p<p1pp1pp1>
1133212223

Code: Select all

`* * *****  * * * *  *   * *`
$\mathfrak{p}(\omega^{\omega2+1}+\omega^{\omega}+1)$

Prime: No
Symmetrical: No
Alphabetical: No

Length: 20
Reversals: 9
Max Depth: 3
Factors: 3
Smoothness: 61
Necessary Brackets: 2

My script now includes emlightened's ordinal notation (hopefully it's correct – I don't know enough about ordinals to be sure)