Count up with recursive prime factorization

[tomtom2357]

691
<<<<>>><<<>>><<<>>>>
<ccc>

[phillip1882]

692
<><><<><><><<<>>>>
72.66% efficiency

[tomtom2357]

693
<<>><<>><<><>><<<<>>>>
bb<aa>d

[phillip1882]

694
<><<<>><<<>><<>>>>
72.69% efficiency

[Elmach]

695
<<<>>><<><<<><>>>>
c<a<<aa>>>
Fp(Pd(PP))
33213114

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tertius et prīmus (prīmī et secundī bisprīmī)

7 by 5 with four extra dots; not happy with how the image turned out at the bottom right of the 139.

695bl.png (3.47 KiB) Viewed 2214 times

Spoiler:

A little happier with this version,

695bl2.png (3.44 KiB) Viewed 2212 times

or maybe this one.

695bl3.png (3.45 KiB) Viewed 2209 times

A version with arms

695b5al.png (1.56 KiB) Viewed 2206 times

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

Nodes: 9
Reversals: 7
Max Depth: 4
Smoothness: 139

[tomtom2357]

696
<><><><<>><<><<<>>>>
aaab<ac>

[Elmach]

697
<<<><>>><<<><<>>>>
<<aa>><<ab>>
d(PP)d(PD)
31133124

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secundus bisprīmī et secundus (prīmī et secundī)

8 by 4 with one dot extra

697al.png (1.43 KiB) Viewed 2193 times

Semiprime
Asymmetric
Not Alphabetic
Square-free, but recursively only cube-free

Nodes: 9
Reversals: 7
Max Depth: 4
Smoothness: 41

[tomtom2357]

698
<><<>><<<<>><<>>>>
ab<<bb>>

[Elmach]

699
<<>><<<>><<<><>>>>
b<b<<aa>>>
Dp(Dd(PP))
22323114

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secundus et prīmus (secundī et secundī bisprīmī)

6 by 5 exactly.

699l.png (1.59 KiB) Viewed 2182 times

Semiprime
Asymmetric
Not Alphabetic
Square-free, but recursively only cube-free

Nodes: 9
Reversals: 7
Max Depth: 4
Smoothness: 233

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Mistaken labyrinth notation, where I replaced a 3 with a 5, making 1165.

Spoiler:

7 by 6 with seven unused dots, not optimal - EDIT: WRONG - this is 1165.

699l.png (1.67 KiB) Viewed 2191 times

[phillip1882]

700
<><><<<>>><<<>>><<><>>
59.56% efficiency

[Elmach]

701
<<><<>><<>><<><>>>
<abb<aa>>
p(PDDp(PP))
2122222113

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prīmus (prīmī et bissecundī et prīmī bisprīmī)

5 by 5 with one extra dot.

701l.png (1.36 KiB) Viewed 2170 times

Prime
Asymmetric
Not Alphabetic
Square-free, but recursively only cube-free

Nodes: 9
Reversals: 9
Max Depth: 3

[tomtom2357]

702
<><<>><<>><<>><<><<>>>
abbb<ab>

[Elmach]

703
<<><><>><<><><<>>>
<aaa><aab>
p(PPP)p(PPD)
211112211123

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prīmus terprīmī et prīmus (bisprīmī et secundī)

5 by 4 exactly...

703al.png (1.27 KiB) Viewed 2162 times

but if I arrange it like this, you can see how similar the two factors are.

703bl.png (1.42 KiB) Viewed 2162 times

Semiprime (19 * 37)
Asymmetric
Not Alphabetic
Square-free, but recursively only tetradbiquadrate-free

Nodes: 9
Reversals: 11
Max Depth: 3
Smoothness: 37

[tomtom2357]

704
<><><><><><><<<<>>>>
aaaaaad

[phillip1882]

705
<<>><<<>>><<<>><<<>>>>
59.62% efficiency

[tomtom2357]

706
<><<<><><<<>>>>>
a<<aac>>

[Elmach]

707
<<><>><<><<><<>>>>
<aa><a<ab>>
spPPpPpPD¹ or p(PP)p(Pp(Pp(P)))¹ or spPPpPpPpPpPS¹
2112212124

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prīmus bisprīmī et prīmus (prīmī et prīmī (prīmī et secundī))

7 by 4 with two extra dots.

707al.png (1.45 KiB) Viewed 2157 times

Semiprime (7 * 101)
Asymmetric
Not Alphabetic
Square-free, but recursively only cube-free

Nodes: 9
Reversals: 9
Max Depth: 4
Smoothness: 101
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¹Notes on this version of p-notation I've been developing

Spoiler:

It occurs to me that I've been using this notation for a while, and haven't really defined it anywhere (I think.). s, p, d, f, g, etc., are prefix binary functions (p*q, p*q^th prime, etc.) and S, P, D, F, G, etc., are constants. (1 = "", 2 = <>, 3 = <<>>, 5 = <<<>>>, 11 = <<<<>>>>, etc..) This way, we can have a nonambiguous grammar without parentheses.

Of course, if I put parentheses afterwards, then the function just applies to the value in front, so s(Number) evaluates to Number, p(Number) evaluates to Number^th prime, etc.; also, in this case, assume sequences of terms evaluate to the product.

Without s, S, or parentheses, we can represent any prime with recursively no more than two factors. With parentheses, or with s, we can represent any number (where 1 is represented by the empty string).

[tomtom2357]

*sigh* We all want to have 709 don't we?
708
<><><<>><<<><>>>

[Elmach]

I just realized that 709 is a special number. That explains why the activity suddenly dropped.
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709
<<<<<<<>>>>>>>
g (alphabet)
J (SPDFGHIJ/orbitals)
77 (7 up, then 7 down)

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septimus

7 by 7.

709l.png (3.79 KiB) Viewed 2136 times

bad attempt of making circles

709al.png (4.51 KiB) Viewed 2133 times

Recursively prime/alphabetic atom. This sequence goes 1,2,3,5,11,31,127,709. The next one is 5381. According to OEIS, this is A007097.
Nodes: 7
Reversals: 1
Max Depth: 7
----