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recursive number addition


phil1882
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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?

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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)

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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.

<<><>>

 

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