FRACTRAN is a Turing-complete esoteric programming language invented by the mathematician John Horton Conway.

A FRACTRAN program is an ordered list of positive fractions 



P
=
(

f

1


,

f

2


,
…
,

f

m


)


{\displaystyle P=(f_{1},f_{2},\ldots ,f_{m})}

, together with an initial positive integer input 



n


{\displaystyle n}

.



The program is run by updating the integer 



n


{\displaystyle n}

 as follows:

 for the first fraction, 




f

i




{\displaystyle f_{i}}

, in the list for which 



n

f

i




{\displaystyle nf_{i}}

 is an integer, replace 



n


{\displaystyle n}

 with 



n

f

i




{\displaystyle nf_{i}}

 ;
 repeat this rule until no fraction in the list produces an integer when multiplied by 



n


{\displaystyle n}

, then halt.


Conway gave a program for primes in FRACTRAN:

 



17

/

91


{\displaystyle 17/91}

, 



78

/

85


{\displaystyle 78/85}

, 



19

/

51


{\displaystyle 19/51}

, 



23

/

38


{\displaystyle 23/38}

, 



29

/

33


{\displaystyle 29/33}

, 



77

/

29


{\displaystyle 77/29}

, 



95

/

23


{\displaystyle 95/23}

, 



77

/

19


{\displaystyle 77/19}

, 



1

/

17


{\displaystyle 1/17}

, 



11

/

13


{\displaystyle 11/13}

, 



13

/

11


{\displaystyle 13/11}

, 



15

/

14


{\displaystyle 15/14}

, 



15

/

2


{\displaystyle 15/2}

, 



55

/

1


{\displaystyle 55/1}


Starting with 



n
=
2


{\displaystyle n=2}

, this FRACTRAN program will change 



n


{\displaystyle n}

 to 



15
=
2
×
(
15

/

2
)


{\displaystyle 15=2\times (15/2)}

, then 



825
=
15
×
(
55

/

1
)


{\displaystyle 825=15\times (55/1)}

, generating the following sequence of integers:

 



2


{\displaystyle 2}

, 



15


{\displaystyle 15}

, 



825


{\displaystyle 825}

, 



725


{\displaystyle 725}

, 



1925


{\displaystyle 1925}

, 



2275


{\displaystyle 2275}

, 



425


{\displaystyle 425}

, 



390


{\displaystyle 390}

, 



330


{\displaystyle 330}

, 



290


{\displaystyle 290}

, 



770


{\displaystyle 770}

, 



…


{\displaystyle \ldots }


After 2, this sequence contains the following powers of 2:






2

2


=
4


{\displaystyle 2^{2}=4}

, 




2

3


=
8


{\displaystyle 2^{3}=8}

, 




2

5


=
32


{\displaystyle 2^{5}=32}

, 




2

7


=
128


{\displaystyle 2^{7}=128}

, 




2

11


=
2048


{\displaystyle 2^{11}=2048}

, 




2

13


=
8192


{\displaystyle 2^{13}=8192}

, 




2

17


=
131072


{\displaystyle 2^{17}=131072}

, 




2

19


=
524288


{\displaystyle 2^{19}=524288}

, 



…


{\displaystyle \ldots }


which are the prime powers of 2.



Task

Write a program that reads a list of fractions in a natural format from the keyboard or from a string,
to parse it into a sequence of fractions (i.e. two integers),
and runs the FRACTRAN starting from a provided integer, writing the result at each step.
It is also required that the number of steps is limited (by a parameter easy to find).



Extra credit

Use this program to derive the first 20 or so prime numbers.



See also

For more on how to program FRACTRAN as a universal programming language, see:

 J. H. Conway (1987). Fractran: A Simple Universal Programming Language for Arithmetic. In: Open Problems in Communication and Computation, pages 4–26. Springer.
 J. H. Conway (2010). "FRACTRAN: A simple universal programming language for arithmetic". In Jeffrey C. Lagarias. The Ultimate Challenge: the 3x+1 problem. American Mathematical Society. pp. 249–264. ISBN 978-0-8218-4940-8. Zbl 1216.68068.
 Number Pathology: Fractran by Mark C. Chu-Carroll; October 27, 2006.


with Ada.Text_IO;
 
procedure Fractan is
 
   type Fraction is record Nom: Natural; Denom: Positive; end record;
   type Frac_Arr is array(Positive range <>) of Fraction;
 
   function "/" (N: Natural; D: Positive) return Fraction is
      Frac: Fraction := (Nom => N, Denom => D);
   begin
      return Frac;
   end "/";
 
   procedure F(List: Frac_Arr; Start: Positive; Max_Steps: Natural) is
      N: Positive := Start;
      J: Positive;
   begin
      Ada.Text_IO.Put(" 0:" & Integer'Image(N) & "   ");
      for I in 1 .. Max_Steps loop
	 J := List'First;
	 loop
	    if N mod List(J).Denom = 0 then
	       N := (N/List(J).Denom) * List(J).Nom;
	       exit; -- found fraction
	    elsif J >= List'Last then
	       return; -- did try out all fractions
	    else
	       J := J + 1; -- try the next fraction
	    end if;
	 end loop;
	 Ada.Text_IO.Put(Integer'Image(I) & ":" & Integer'Image(N) & "   ");
      end loop;
   end F;
 
begin
   -- F((2/3, 7/2, 1/5, 1/7, 1/9, 1/4, 1/8), 2, 100); 
   -- output would be "0: 2    1: 7    2: 1" and then terminate
 
   F((17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 
      77/19,  1/17, 11/13, 13/11, 15/14,  15/2, 55/1), 
     2, 15);
   -- output is "0: 2    1: 15    2: 825    3: 725   ...   14: 132    15: 116"
end Fractan;