recursive number addition

[phil1882]

if n is composite, break it into its prime factors, if n is prime, place < > around n and replace n with the nth prime its is. the first 20 numbers then are...
1

<> 2

<<>> 3

<><> 4

<<<>>> 5

<><<>> 6

<<><>> 7

<><><> 8

<<>><<>> 9

<><<<>>> 10

<<<<>>>> 11

<><><<>> 12

<<><<>>> 13

<><<><>>14

<<>><<<>>> 15

<><><><> 16

<<<><>>> 17

<><<>><<>> 18

<<><><>> 19

<><><<<>>> 20

hypothesis:

1) there is no general method for adding recursive numbers.

2) numbers that differ by 1 wont differ in recursive representation by more than 2 brackets

3) symmetric recursive numbers, recursive numbers that can be represented in such a way that they are a mirror image at the middle, grow at a logarithmic rate somewhat similar to the primes.

can you confirm or disprove any of these?

[bonanova]

I'm not getting this. Can you do the first couple numbers step by step?

How to you decided on n? Just starting with 1, 2, 3, ... ?

Where does "addition" come in?

Thx.

[CaptainEd]

Here's my guess about the rules. I have no answers for your questions, Bonanova.

S(i) is the string for number i.
As I read the examples, I infer that
* S(1) is the empty string
* if j is non-prime, replace j with the string of its factors (e.g. replace 12 with 2 2 3)
* replace the i-th prime with angle brackets around S(i) (counting 2 as the 1-th prime)

[phil1882]

correct.

 

one of my questions is about addition.

basically my question is, given a number system that uses multiplcation as its basis, (multiplying two numbers in recursive format is just concatenation) an you come up with a method for addition?

ill do 7 as an example

7 is prime.

< 7 >

7 is the 4th prime so replace the 7 with a 4.

< 4 >

4 is composite, its factors are 2 and 2.

< 2 2 >

2 and 2 are prime.

<<2><2>>

the number two is the first prime so cant be reduced further. replace with blank.

<<><>>

 

[CaptainEd]

Are you sure multiplication is only concatenation? 4 * 3 and 3 * 4 are different strings, and only 4*3 looks like S(12).

[phil1882]

the strings <<>><><>, <><><<>> are considered equivalent, though for the proposes of uniqueness, i put the factors in order from lowest to highest.

[jasen]

I have create a program to count braces from 1 to 5000

1. I don't know what you mean "recursive numbers".

2. I found hypothesis 2 is wrong
   4999 = <<<>><<><><><><<>>>>  = 10 brackets
   5000 = <><><><<<>>><<<>>><<<>>><<<>>> = 15 brackets

Edited November 20, 2015 by jasen

[phil1882]

alright thanks good to know.