# Fractran

Fractran
You are encouraged to solve this task according to the task description, using any language you may know.

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 ${\displaystyle P=(f_{1},f_{2},\ldots ,f_{m})}$, together with an initial positive integer input ${\displaystyle n}$.

The program is run by updating the integer ${\displaystyle n}$ as follows:

• for the first fraction, ${\displaystyle f_{i}}$, in the list for which ${\displaystyle nf_{i}}$ is an integer, replace ${\displaystyle n}$ with ${\displaystyle nf_{i}}$ ;
• repeat this rule until no fraction in the list produces an integer when multiplied by ${\displaystyle n}$, then halt.

Conway gave a program for primes in FRACTRAN:

${\displaystyle 17/91}$, ${\displaystyle 78/85}$, ${\displaystyle 19/51}$, ${\displaystyle 23/38}$, ${\displaystyle 29/33}$, ${\displaystyle 77/29}$, ${\displaystyle 95/23}$, ${\displaystyle 77/19}$, ${\displaystyle 1/17}$, ${\displaystyle 11/13}$, ${\displaystyle 13/11}$, ${\displaystyle 15/14}$, ${\displaystyle 15/2}$, ${\displaystyle 55/1}$

Starting with ${\displaystyle n=2}$, this FRACTRAN program will change ${\displaystyle n}$ to ${\displaystyle 15=2\times (15/2)}$, then ${\displaystyle 825=15\times (55/1)}$, generating the following sequence of integers:

${\displaystyle 2}$, ${\displaystyle 15}$, ${\displaystyle 825}$, ${\displaystyle 725}$, ${\displaystyle 1925}$, ${\displaystyle 2275}$, ${\displaystyle 425}$, ${\displaystyle 390}$, ${\displaystyle 330}$, ${\displaystyle 290}$, ${\displaystyle 770}$, ${\displaystyle \ldots }$

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

${\displaystyle 2^{2}=4}$, ${\displaystyle 2^{3}=8}$, ${\displaystyle 2^{5}=32}$, ${\displaystyle 2^{7}=128}$, ${\displaystyle 2^{11}=2048}$, ${\displaystyle 2^{13}=8192}$, ${\displaystyle 2^{17}=131072}$, ${\displaystyle 2^{19}=524288}$, ${\displaystyle \ldots }$

which are the prime powers of 2.

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.

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.

## 11l

Translation of: D
F fractran(prog, =val, limit)
V fracts = prog.split(‘ ’).map(p -> p.split(‘/’).map(i -> Int(i)))
[Float] r
L(n) 0 .< limit
r [+]= val
L(p) fracts
I val % p[1] == 0
val = p[0] * val / p[1]
L.break
R r

print(fractran(‘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:
[2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132]


## 360 Assembly

*        FRACTRAN                  17/02/2019
FRACTRAN CSECT
USING  FRACTRAN,R13       base register
B      72(R15)            skip savearea
DC     17F'0'             savearea
SAVE   (14,12)            save previous context
LA     R6,1               i=1
DO WHILE=(C,R6,LE,TERMS)    do i=1 to terms
LA     R7,1                 j=1
DO WHILE=(C,R7,LE,=A(NF))     do j=1 to nfracs
LR     R1,R7                  j
SLA    R1,3                   ~
L      R2,FRACS-4(R1)         d(j)
L      R4,NN                  nn
SRDA   R4,32                  ~
DR     R4,R2                  nn/d(j)
IF       LTR,R4,Z,R4 THEN       if mod(nn,d(j))=0 then
XDECO  R6,XDEC                  edit i
MVC    PG(3),XDEC+9             output i
L      R1,NN                    nn
XDECO  R1,PG+5                  edit & output nn
XPRNT  PG,L'PG                  print buffer
LR     R1,R7                    j
SLA    R1,3                     ~
L      R3,FRACS-8(R1)           n(j)
MR     R4,R3                    *n(j)
ST     R5,NN                    nn=nn/d(j)*n(j)
B      LEAVEJ                   leave j
ENDIF    ,                      end if
LA     R7,1(R7)               j++
ENDDO    ,                    end do j
LEAVEJ   LA     R6,1(R6)             i++
ENDDO    ,                  end do i
L      R13,4(0,R13)       restore previous savearea pointer
RETURN (14,12),RC=0       restore registers from calling sav
NF       EQU    (TERMS-FRACS)/8    number of fracs
NN       DC     F'2'               nn
FRACS  DC  F'17',F'91',F'78',F'85',F'19',F'51',F'23',F'38',F'29',F'33'
DC  F'77',F'29',F'95',F'23',F'77',F'19',F'1',F'17',F'11',F'13'
DC  F'13',F'11',F'15',F'14',F'15',F'2',F'55',F'1'
TERMS    DC     F'100'             terms
PG       DC     CL80'*** :'        buffer
XDEC     DS     CL12               temp
REGEQU
END    FRACTRAN
Output:
  1 :           2
2 :          15
3 :         825
4 :         725
5 :        1925
6 :        2275
7 :         425
8 :         390
9 :         330
10 :         290
...
99 :        2128
100 :        1288


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;

Output:
 0: 2    1: 15    2: 825    3: 725    4: 1925    5: 2275    6: 425    7: 390    8: 330    9: 290    10: 770    11: 910    12: 170    13: 156    14: 132    15: 116

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.win32
# as the numbers required for finding the first 20 primes are quite large, #
# we use Algol 68G's LONG LONG INT with a precision of 100 digits          #
PR precision 100 PR

# mode to hold fractions #
MODE FRACTION = STRUCT( INT numerator, INT denominator );

# define / between two INTs to yield a FRACTION #
OP / = ( INT a, b )FRACTION: ( a, b );

# mode to define a FRACTRAN progam #
MODE FRACTRAN = STRUCT( FLEX[0]FRACTION data
, LONG LONG INT   n
, BOOL            halted
);
# prepares a FRACTRAN program for use - sets the initial value of n and halted to FALSE #
PRIO STARTAT = 1;
OP   STARTAT = ( REF FRACTRAN f, INT start )REF FRACTRAN:
BEGIN
halted OF f := FALSE;
n OF f := start;
f
END;

# sets n OF f to the next number in the sequence or sets halted OF f to TRUE if the sequence has ended #
OP NEXT = ( REF FRACTRAN f )LONG LONG INT:
IF halted OF f
THEN n OF f := 0
ELSE
BOOL          found  := FALSE;
LONG LONG INT result := 0;
FOR pos FROM LWB data OF f TO UPB data OF f WHILE NOT found DO
LONG LONG INT value       = n OF f * numerator OF ( ( data OF f )[ pos ] );
INT           denominator = denominator OF ( ( data OF f )[ pos ] );
IF found := ( value MOD denominator = 0 ) THEN result := value OVER denominator FI
OD;
n OF f := result
FI ;

# generate and print the sequence of numbers from a FRACTRAN pogram #
PROC print fractran sequence = ( REF FRACTRAN f, INT start, INT limit )VOID:
BEGIN
VOID( f STARTAT start );
print( ( "0: ", whole( start, 0 ) ) );
FOR i TO limit
WHILE VOID( NEXT f );
NOT halted OF f
DO
print( ( " " + whole( i, 0 ) + ": " + whole( n OF f, 0 ) ) )
OD;
print( ( newline ) )
END ;

# print the first 16 elements from the primes FRACTRAN program #
FRACTRAN pf := ( ( 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 ), 0, FALSE );
print fractran sequence( pf, 2, 15 );

# find some primes using the pf FRACTRAN progam - n is prime for the members in the sequence that are 2^n #
INT primes found := 0;
VOID( pf STARTAT 2 );
INT pos := 0;
print( ( "seq position  prime sequence value", newline ) );
WHILE primes found < 20 AND NOT halted OF pf DO
LONG LONG INT value      := NEXT pf;
INT      power of 2 := 0;
pos +:= 1;
WHILE value MOD 2 = 0 AND value > 0 DO power of 2 PLUSAB 1; value OVERAB 2 OD;
IF value = 1 THEN
# found a prime #
primes found +:= 1;
print( ( whole( pos, -12 ) + " " + whole( power of 2, -6 ) + " (" + whole( n OF pf, 0 ) + ")", newline ) )
FI
OD
Output:
0: 2 1: 15 2: 825 3: 725 4: 1925 5: 2275 6: 425 7: 390 8: 330 9: 290 10: 770 11: 910 12: 170 13: 156 14: 132 15: 116
seq position  prime sequence value
19      2 (4)
69      3 (8)
280      5 (32)
707      7 (128)
2363     11 (2048)
3876     13 (8192)
8068     17 (131072)
11319     19 (524288)
19201     23 (8388608)
36866     29 (536870912)
45551     31 (2147483648)
75224     37 (137438953472)
101112     41 (2199023255552)
117831     43 (8796093022208)
152025     47 (140737488355328)
215384     53 (9007199254740992)
293375     59 (576460752303423488)
327020     61 (2305843009213693952)
428553     67 (147573952589676412928)
507519     71 (2361183241434822606848)


## APL

Works with: Dyalog APL
fractran←{
parts ← ' '∘≠⊆⊢
frac ← ⍎¨'/'∘≠⊆⊢
simp ← ⊢÷∨/
mul ← simp×
prog ← simp∘frac¨parts ⍺
step ← {⊃⊃(1=2⊃¨next)/next←⍺ mul¨⊂(⍵ 1)}
(start nstep)←⍵
rslt ← ⊃(⊢,⍨prog∘step∘⊃)⍣nstep¨start
⌽(⊢(/⍨)(∨\0∘≠))rslt
}

Output:
      '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' fractran 2 20
2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4 30


## AutoHotkey

n := 2, steplimit := 15, numerator := [], denominator := []
s := "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"

Loop, Parse, s, % A_Space
if (!RegExMatch(A_LoopField, "^(\d+)/(\d+)$", m)) MsgBox, % "Invalid input string (" A_LoopField ")." else numerator[A_Index] := m1, denominator[A_Index] := m2 SetFormat, FloatFast, 0.0 Gui, Add, ListView, R10 W100 -Hdr, | SysGet, VSBW, 2 LV_ModifyCol(1, 95 - VSBW), LV_Add( , 0 ": " n) Gui, Show Loop, % steplimit { i := A_Index Loop, % numerator.MaxIndex() if (!Mod(nn := n * numerator[A_Index] / denominator[A_Index], 1)) { LV_Modify(LV_Add( , i ": " (n := nn)), "Vis") continue, 2 } break }  Output: 0: 2 1: 15 2: 825 3: 725 4: 1925 5: 2275 6: 425 7: 390 8: 330 9: 290 10: 770 11: 910 12: 170 13: 156 14: 132 15: 116 ## bash Works with: Bash the "factor" command allows one to decrypt the data. For example, the program below computes the product of a and b, entered as 2a and 3b, the product being 5a×b. Two arrays are computed from the fractions, ns for the numerators and ds for the denominators. Then, every time where the multiplication by a fraction yields an integer, the output of the division is stored into a csv file in factored format. #! /bin/bash program="1/1 455/33 11/13 1/11 3/7 11/2 1/3" echo$program | tr " " "\n" | cut -d"/" -f1 | tr "\n" " " > "data"
echo $program | tr " " "\n" | cut -d"/" -f2 | tr "\n" " " > "data" read -a ds < "data" t=0 n=72 echo "steps of computation" > steps.csv while [$t -le 6 ]; do
if [ $(($n*${ns[$t]}%${ds[$t]})) -eq 0 ]; then
let "n=$(($n*${ns[$t]}/${ds[$t]}))"
let "t=0"
factor $n >> steps.csv fi let "t=$t+1"
done


If at the beginning n=72=23×32 (to compute 3×2), the steps of the computation look like this:

steps of computation
72: 2 2 2 3 3
396: 2 2 3 3 11
5460: 2 2 3 5 7 13
4620: 2 2 3 5 7 11
63700: 2 2 5 5 7 7 13
53900: 2 2 5 5 7 7 11
4900: 2 2 5 5 7 7
2100: 2 2 3 5 5 7
900: 2 2 3 3 5 5
4950: 2 3 3 5 5 11
68250: 2 3 5 5 5 7 13
57750: 2 3 5 5 5 7 11
796250: 2 5 5 5 5 7 7 13
673750: 2 5 5 5 5 7 7 11
61250: 2 5 5 5 5 7 7
26250: 2 3 5 5 5 5 7
11250: 2 3 3 5 5 5 5
61875: 3 3 5 5 5 5 11
853125: 3 5 5 5 5 5 7 13
721875: 3 5 5 5 5 5 7 11
9953125: 5 5 5 5 5 5 7 7 13
8421875: 5 5 5 5 5 5 7 7 11
765625: 5 5 5 5 5 5 7 7
328125: 3 5 5 5 5 5 5 7
140625: 3 3 5 5 5 5 5 5
46875: 3 5 5 5 5 5 5
15625: 5 5 5 5 5 5


This file can be opened with a spreadsheet to draw the successive states of the prime numbers (with countif) and then look how the computation is done in successive steps.

## Batch File

@echo off
setlocal enabledelayedexpansion

::Set the inputs
set "code=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"
set "n=2"

::Basic validation of code
for %%. in (!code!) do (
echo.%%.|findstr /r /c:"^[0-9][0-9]*/[1-9][0-9]*$">nul||goto error_code ) ::Validate the input set /a "tst=1*!n!" 2>nul if !tst! lss 0 goto error_input if !tst! equ 0 (if not "!n!"=="0" (goto error_input)) ::Set the limit outputs set limit=20 ::Execute the code echo.Input: echo. !n! echo.Output: for /l %%? in (1,1,!limit!) do ( set shouldwehalt=1 for %%A in (!code!) do ( for /f "tokens=1,2 delims=/" %%B in ("%%A") do ( set /a "tst=!n! %% %%C" if !tst! equ 0 ( if !shouldwehalt! equ 1 ( set shouldwehalt=0 set /a "n=n*%%B/%%C" echo. !n! ) ) ) ) if !shouldwehalt! equ 1 goto halt ) :halt echo. pause exit /b 0 :error_code echo.Syntax error in code. echo. pause exit /b 1 :error_input echo.Invalid input. echo. pause exit /b 1  Output: Input: 2 Output: 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4 30 Press any key to continue . . . ## Befunge This takes as input a space-separated list of fractions, a starting value, and the number of iterations to output. Note that in some interpreters you may need to press <Return> twice after entering the fractions if the Starting value prompt doesn't at first appear. p0" :snoitcarF">:#,_>&00g5p~$&00g:v
v"Starting value: "_^#-*84~p6p00+1<
>:#,_&0" :snoitaretI">:#,_#@>>$&\:v :$_\:10g5g*:10g6g%v1:\1$\$<|!:-1\.<
g0^<!:-1\p01+1g01$_10g6g/\^>\010p00  Output: Fractions: 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 Starting value: 2 Iterations: 16 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 ## BQN The function RunFractran runs a fractran program, given max iterations on the left, and input, program string on the right. It returns a list of generated numbers. Fractran performs a single iteration of fractran on a given input, list of numerators and list of denominators. # Fractran interpreter # Helpers _while_ ← {𝔽⍟𝔾∘𝔽_𝕣_𝔾∘𝔽⍟𝔾𝕩} ToInt ← 10⊸×⊸+˜´·⌽-⟜'0' ToFrac ← { i ← ⊑/'/'=𝕩 ToInt¨i(↑⋈1⊸+⊸↓)𝕩 } Split ← ((¬-˜⊢×·+»⊸>)∘≠⊔⊢) Fractran ← { 𝕊 n‿num‿den: ind ← ⊑/0=den|num×n ⟨(n×ind⊑num)÷ind⊑den ⋄ num ⋄ den⟩ } RunFractran ← { steps 𝕊 inp‿prg: num‿den ← <˘⍉>ToFrac¨' 'Split prg step ← 1 list ← ⟨inp⟩ { step +↩ 1 out ← Fractran 𝕩 list ∾↩ ⊑out out } _while_ {𝕊 n‿num‿den: (step<steps)∧ ∨´0=den|num} inp‿num‿den list } seq ← 200 RunFractran 2‿"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" •Out "Generated numbers: "∾•Repr seq •Out "Primes: "∾•Repr 1↓⌊2⋆⁼(⌈=⌊)∘(2⊸(⋆⁼))⊸/ seq   )ex fractran.bqn Generated numbers: 2‿15‿825‿725‿1925‿2275‿425‿390‿330‿290‿770‿910‿170‿156‿132‿116‿308‿364‿68‿4‿30‿225‿12375‿10875‿28875‿25375‿67375‿79625‿14875‿13650‿2550‿2340‿1980‿1740‿4620‿4060‿10780‿12740‿2380‿2184‿408‿152‿92‿380‿230‿950‿575‿2375‿9625‿11375‿2125‿1950‿1650‿1450‿3850‿4550‿850‿780‿660‿580‿1540‿1820‿340‿312‿264‿232‿616‿728‿136‿8‿60‿450‿3375‿185625‿163125‿433125‿380625‿1010625‿888125‿2358125‿2786875‿520625‿477750‿89250‿81900‿15300‿14040‿11880‿10440‿27720‿24360‿64680‿56840‿150920‿178360‿33320‿30576‿5712‿2128‿1288‿5320‿3220‿13300‿8050‿33250‿20125‿83125‿336875‿398125‿74375‿68250‿12750‿11700‿9900‿8700‿23100‿20300‿53900‿63700‿11900‿10920‿2040‿1872‿1584‿1392‿3696‿3248‿8624‿10192‿1904‿112‿120‿900‿6750‿50625‿2784375‿2446875‿6496875‿5709375‿15159375‿13321875‿35371875‿31084375‿82534375‿97540625‿18221875‿16721250‿3123750‿2866500‿535500‿491400‿91800‿84240‿71280‿62640‿166320‿146160‿388080‿341040‿905520‿795760‿2112880‿2497040‿466480‿428064‿79968‿29792‿18032‿74480‿45080‿186200‿112700‿465500‿281750‿1163750‿704375‿2909375‿11790625‿13934375‿2603125‿2388750‿446250‿409500‿76500‿70200‿59400‿52200‿138600‿121800‿323400‿284200‿754600‿891800‿166600‿152880‿28560‿26208‿4896‿1824‿1104 Primes: 2‿3  ## Bracmat This program computes the first twenty primes. It has to do almost 430000 iterations to arrive at the twentieth prime, so instead of immediately writing each number to the terminal, it adds it to a list. After the set number of iterations, the list of numbers is written to a text file numbers.lst (21858548 bytes), so you can inspect it. Because it takes some time to do all iterations, its is advisable to write the source code below in a text file 'fractran' and run it in batch mode in the background, instead of starting Bracmat in interactive mode and typing the program at the prompt. The primes, together with the largest number found, are written to a file FRACTRAN.OUT. (fractran= np n fs A Z fi P p N L M . !arg:(?N,?n,?fs) {Number of iterations, start n, fractions} & :?P:?L {Initialise accumulators.} & whl ' ( -1+!N:>0:?N {Stop when counted down to zero.} & !n !L:?L {Prepend all numbers to result list.} & (2\L!n:#?p&!P !p:?P|) {If log2(n) is rational, append it to list of primes.} & !fs:? (/?fi&!n*!fi:~/:?n) ? {This line does the following (See task description): "for the first fraction, fi, in the list for which nfi is an integer, replace n by nfi ;"} ) & :?M & whl'(!L:%?n ?L&!n !M:?M) {Invert list of numbers. (append to long list is very expensive. Better to prepend and finally invert.} & (!M,!P) {Return the two lists} ); ( clk$:?t0
& fractran$(430000, 2, 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) : (?numbers,?primes) & lst$(numbers,"numbers.lst",NEW)
& put$(" FRACTRAN found these primes:" !primes "\nThe list of numbers is saved in numbers.txt The biggest number in the list is" ( 0:?max & !numbers:? (>%@!max:?max&~) ? | !max ) str$("\ntime: " flt$(clk$+-1*!t0,4) " sec\n")
, "FRACTRAN.OUT",NEW)
);

In Linux, run the program as follows (assuming bracmat and the file 'fractran' are in the CWD):

./bracmat 'get$fractran' & Output: in FRACTRAN.OUT FRACTRAN found these primes: 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 The list of numbers is saved in numbers.txt The biggest number in the list is 1842775069354845065175076326808495219647033145169559640049770986129640260031692378106527030467987060546875 time: 1,8668*10E3 sec  ## C Using GMP. Powers of two are in brackets. For extra credit, pipe the output through | less -S. #include <stdio.h> #include <stdlib.h> #include <gmp.h> typedef struct frac_s *frac; struct frac_s { int n, d; frac next; }; frac parse(char *s) { int offset = 0; struct frac_s h = {0}, *p = &h; while (2 == sscanf(s, "%d/%d%n", &h.n, &h.d, &offset)) { s += offset; p = p->next = malloc(sizeof *p); *p = h; p->next = 0; } return h.next; } int run(int v, char *s) { frac n, p = parse(s); mpz_t val; mpz_init_set_ui(val, v); loop: n = p; if (mpz_popcount(val) == 1) gmp_printf("\n[2^%d = %Zd]", mpz_scan1(val, 0), val); else gmp_printf(" %Zd", val); for (n = p; n; n = n->next) { // assuming the fractions are not reducible if (!mpz_divisible_ui_p(val, n->d)) continue; mpz_divexact_ui(val, val, n->d); mpz_mul_ui(val, val, n->n); goto loop; } gmp_printf("\nhalt: %Zd has no divisors\n", val); mpz_clear(val); while (p) { n = p->next; free(p); p = n; } return 0; } int main(void) { run(2, "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"); return 0; }  ## C# The use of  using Fractype = (BigInteger numerator, BigInteger denominator); requires C# 12.0. namespace System.Numerics { using Fractype = (BigInteger numerator, BigInteger denominator); struct Quotient { private Fractype _frac; public Fractype Fraction { get => _frac; set => _frac = Reduce(value); } public bool IsIntegral => _frac.denominator == 1; public Quotient(BigInteger num, BigInteger den) { Fraction = (num, den); } public static BigInteger GCD(BigInteger a, BigInteger b) { return (b == 0) ? a : GCD(b, a % b); } private static Fractype Reduce(Fractype f) { if (f.denominator == 0) throw new DivideByZeroException(); BigInteger gcd = Quotient.GCD(f.numerator, f.denominator); return (f.numerator / gcd, f.denominator / gcd); } public static Quotient operator *(Quotient a, Quotient b) => new Quotient(a._frac.numerator * b._frac.numerator, a._frac.denominator * b._frac.denominator); public static Quotient operator *(Quotient a, BigInteger n) => new Quotient(a._frac.numerator * n, a._frac.denominator); public static explicit operator Quotient(Fractype t) => new Quotient(t.numerator, t.denominator); } class FRACTRAN { private Quotient[] code; public FRACTRAN(Fractype[] _code) { code = _code.Select(x => (Quotient) x).ToArray(); } public (BigInteger value, bool success) Compute(BigInteger n) { for (int i = 0; i < code.Length; i++) if ((code[i] * n).IsIntegral) return ((code[i] * n).Fraction.numerator, true); return (0, false); } } class Program { public static void Main(string[] args) { Fractype[] frac_code = args[0].Split(" ") .Select(x => ((BigInteger)Int32.Parse(x.Split("/")[0]), (BigInteger)Int32.Parse(x.Split("/")[1].Trim(',')))).ToArray(); BigInteger init = new BigInteger(Int32.Parse(args[1].Trim(','))); int steps = Int32.Parse(args[2].Trim(',')); FRACTRAN FRACGAME = new FRACTRAN(frac_code); List<BigInteger> sequence = new List<BigInteger>(); sequence.Add(init); bool halt = false; for (int i = 0; i < steps - 1; i++) { var k = FRACGAME.Compute(sequence[sequence.Count - 1]); if (k.success) sequence.Add(k.value); else { halt = true; break; } } for (int i = 0; i < sequence.Count; i++) Console.WriteLine((i + 1).ToString() + ": " + sequence[i]); if (halt) Console.WriteLine("HALT"); } } }  Input:  "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: 1: 2 2: 15 3: 825 4: 725 5: 1925 6: 2275 7: 425 8: 390 9: 330 10: 290 11: 770 12: 910 13: 170 14: 156 15: 132  Moreover, modifying the class Program to,  class Program { private static bool PowerOfTwo(BigInteger b) { while (b % 2 == 0) b /= 2; return b == 1; } private static BigInteger BigLog2(BigInteger b) { BigInteger r = 0; while (b > 1) { r++; b /= 2; } return r; } public static void Main(string[] args) { Fractype[] frac_code = args[0].Split(" ") .Select(x => ((BigInteger)Int32.Parse(x.Split("/")[0]), (BigInteger)Int32.Parse(x.Split("/")[1].Trim(',')))).ToArray(); BigInteger init = new BigInteger(Int32.Parse(args[1].Trim(','))); int steps = Int32.Parse(args[2].Trim(',')); FRACTRAN FRACGAME = new FRACTRAN(frac_code); List<BigInteger> sequence = new List<BigInteger>(); List<BigInteger> primes = new List<BigInteger>(); sequence.Add(init); bool halt = false; while (primes.Count() < 20) { var k = FRACGAME.Compute(sequence[sequence.Count - 1]); if (k.success) sequence.Add(k.value); else { halt = true; break; } if (PowerOfTwo(k.value)) primes.Add(BigLog2(k.value)); } for (int i = 0; i < primes.Count; i++) Console.WriteLine((i + 1).ToString() + ": " + primes[i]); if (halt) Console.WriteLine("HALT"); } }  with the same input, will print the first 20 primes. 1: 2 2: 3 3: 5 4: 7 5: 11 6: 13 7: 17 8: 19 9: 23 10: 29 11: 31 12: 37 13: 41 14: 43 15: 47 16: 53 17: 59 18: 61 19: 67 20: 71  ## C++ #include <iostream> #include <sstream> #include <iterator> #include <vector> #include <cmath> using namespace std; class fractran { public: void run( std::string p, int s, int l ) { start = s; limit = l; istringstream iss( p ); vector<string> tmp; copy( istream_iterator<string>( iss ), istream_iterator<string>(), back_inserter<vector<string> >( tmp ) ); string item; vector< pair<float, float> > v; pair<float, float> a; for( vector<string>::iterator i = tmp.begin(); i != tmp.end(); i++ ) { string::size_type pos = ( *i ).find( '/', 0 ); if( pos != std::string::npos ) { a = make_pair( atof( ( ( *i ).substr( 0, pos ) ).c_str() ), atof( ( ( *i ).substr( pos + 1 ) ).c_str() ) ); v.push_back( a ); } } exec( &v ); } private: void exec( vector< pair<float, float> >* v ) { int cnt = 0; while( cnt < limit ) { cout << cnt << " : " << start << "\n"; cnt++; vector< pair<float, float> >::iterator it = v->begin(); bool found = false; float r; while( it != v->end() ) { r = start * ( ( *it ).first / ( *it ).second ); if( r == floor( r ) ) { found = true; break; } ++it; } if( found ) start = ( int )r; else break; } } int start, limit; }; int main( int argc, char* argv[] ) { fractran f; f.run( "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 ); cin.get(); return 0; }  Output: 0 : 2 1 : 15 2 : 825 3 : 725 4 : 1925 5 : 2275 6 : 425 7 : 390 8 : 330 9 : 290 10 : 770 11 : 910 12 : 170 13 : 156 14 : 132  ## CLU ratio = cluster is new, parse, unparse, get_num, get_denom, mul rep = struct[num, denom: int] new = proc (num, denom: int) returns (cvt) return(simplify(rep${num: num, denom: denom}))
end new

parse = proc (rat: string) returns (ratio) signals (bad_format)
rat := trim(rat)
sep: int := string$indexc('/', rat) if sep = 0 then signal bad_format end num: string := string$substr(rat, 1, sep-1)
denom: string := string$rest(rat, sep+1) return(new(int$parse(num), int$parse(denom))) resignal bad_format end parse trim = proc (s: string) returns (string) start: int := 1 while start <= string$size(s) cand s[start] = ' ' do start := start + 1 end

end_: int := string$size(s) while end_ >= 1 cand s[end_] = ' ' do end_ := end_ - 1 end return(string$substr(s, start, end_-start+1))
end trim

unparse = proc (rat: cvt) returns (string)
return(int$unparse(rat.num) || "/" || int$unparse(rat.denom))
end unparse

get_num = proc (rat: cvt) returns (int)
return(rat.num)
end get_num

get_denom = proc (rat: cvt) returns (int)
return(rat.denom)
end get_denom

mul = proc (a, b: cvt) returns (ratio)
return(new(a.num * b.num, a.denom * b.denom))
end mul

simplify = proc (rat: rep) returns (rep)
num: int := int$abs(rat.num) denom: int := int$abs(rat.denom)

sign: int
if (rat.num < 0) = (rat.denom < 0)
then sign := 1
else sign := -1
end

factor: int := gcd(num, denom)
return(rep${num: sign*num/factor, denom: denom/factor}) end simplify gcd = proc (a, b: int) returns (int) while b ~= 0 do a, b := b, a // b end return(a) end gcd end ratio fractran = cluster is parse, run rep = sequence[ratio] parse = proc (program: string) returns (cvt) parsed: array[ratio] := array[ratio]$[]
for rat: ratio in ratioes(program) do
array[ratio]$addh(parsed, rat) end return(rep$a2s(parsed))
end parse

ratioes = iter (program: string) yields (ratio)
while true do
sep: int := string$indexc(',', program) if sep = 0 then yield(ratio$parse(program))
break
else
yield(ratio$parse(string$substr(program, 1, sep-1)))
program := string$rest(program, sep+1) end end end ratioes run = iter (program: cvt, n, maxiter: int) yields (int) nrat: ratio := ratio$new(n, 1)
while maxiter > 0 do
yield(nrat.num)
begin
for rat: ratio in rep$elements(program) do mul: ratio := rat * nrat if mul.denom = 1 then exit found(mul) end end break end except when found(new: ratio): nrat := new end maxiter := maxiter - 1 end end run end fractran start_up = proc () po: stream := stream$primary_output()

program: string := "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"
parsed: fractran := fractran$parse(program) index: int := 0 for result: int in fractran$run(parsed, 2, 20) do
stream$putright(po, int$unparse(index), 3)
stream$putc(po, ':') stream$putright(po, int$unparse(result), 10) stream$putl(po, "")
index := index + 1
end
end start_up
Output:
  0:         2
1:        15
2:       825
3:       725
4:      1925
5:      2275
6:       425
7:       390
8:       330
9:       290
10:       770
11:       910
12:       170
13:       156
14:       132
15:       116
16:       308
17:       364
18:        68
19:         4

## Common Lisp

(defun fractran (n frac-list)
(lambda ()
(prog1
n
(when n
(let ((f (find-if (lambda (frac)
(integerp (* n frac)))
frac-list)))
(when f (setf n (* f n))))))))

;; test

(defvar *primes-ft* '(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))

(loop with fractran-instance = (fractran 2 *primes-ft*)
repeat 20
for next = (funcall fractran-instance)
until (null next)
do (print next))

Output:
2
15
825
725
1925
2275
425
390
330
290
770
910
170
156
132
116
308
364
68
4

## D

### Simple Version

Translation of: Java
import std.stdio, std.algorithm, std.conv, std.array;

void fractran(in string prog, int val, in uint limit) {
const fracts = prog.split.map!(p => p.split("/").to!(int[])).array;

foreach (immutable n; 0 .. limit) {
writeln(n, ": ", val);
const found = fracts.find!(p => val % p[1] == 0);
if (found.empty)
break;
val = found.front[0] * val / found.front[1];
}
}

void main() {
fractran("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:
0: 2
1: 15
2: 825
3: 725
4: 1925
5: 2275
6: 425
7: 390
8: 330
9: 290
10: 770
11: 910
12: 170
13: 156
14: 132

### Lazy Version

import std.stdio, std.algorithm, std.conv, std.array, std.range;

struct Fractran {
int front;
bool empty = false;
const int[][] fracts;

this(in string prog, in int val) {
this.front = val;
fracts = prog.split.map!(p => p.split("/").to!(int[])).array;
}

void popFront() {
const found = fracts.find!(p => front % p[1] == 0);
if (found.empty)
empty = true;
else
front = found.front[0] * front / found.front[1];
}
}

void main() {
Fractran("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)
.take(15).writeln;
}

Output:
[2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132]

## Delphi

Translation of: Java
program FractranTest;

{$APPTYPE CONSOLE} uses System.SysUtils, System.RegularExpressions; type TFractan = class private limit: Integer; num, den: TArray<Integer>; procedure compile(prog: string); procedure exec(val: Integer); function step(val: Integer): integer; procedure dump(); public constructor Create(prog: string; val: Integer); end; { TFractan } constructor TFractan.Create(prog: string; val: Integer); begin limit := 15; compile(prog); dump(); exec(2); end; procedure TFractan.compile(prog: string); var reg: TRegEx; m: TMatch; begin reg := TRegEx.Create('\s*(\d*)\s*\/\s*(\d*)\s*'); m := reg.Match(prog); while m.Success do begin SetLength(num, Length(num) + 1); num[high(num)] := StrToIntDef(m.Groups[1].Value, 0); SetLength(den, Length(den) + 1); den[high(den)] := StrToIntDef(m.Groups[2].Value, 0); m := m.NextMatch; end; end; procedure TFractan.exec(val: Integer); var n: Integer; begin n := 0; while (n < limit) and (val <> -1) do begin Writeln(n, ': ', val); val := step(val); inc(n); end; end; function TFractan.step(val: Integer): integer; var i: integer; begin i := 0; while (i < length(den)) and (val mod den[i] <> 0) do inc(i); if i < length(den) then exit(round(num[i] * val / den[i])); result := -1; end; procedure TFractan.dump(); var i: Integer; begin for i := 0 to high(den) do Write(num[i], '/', den[i], ' '); Writeln; end; const DATA = '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'; begin TFractan.Create(DATA, 2).Free; Readln; end.  Output: 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 0: 2 1: 15 2: 825 3: 725 4: 1925 5: 2275 6: 425 7: 390 8: 330 9: 290 10: 770 11: 910 12: 170 13: 156 14: 132  ## Elixir Translation of: Erlang defmodule Fractran do use Bitwise defp binary_to_ratio(b) do [_, num, den] = Regex.run(~r/(\d+)\/(\d+)/, b) {String.to_integer(num), String.to_integer(den)} end def load(program) do String.split(program) |> Enum.map(&binary_to_ratio(&1)) end defp step(_, []), do: :halt defp step(n, [f|fs]) do {p, q} = mulrat(f, {n, 1}) case q do 1 -> p _ -> step(n, fs) end end def exec(k, n, program) do exec(k-1, n, fn (_) -> true end, program, [n]) |> Enum.reverse end def exec(k, n, pred, program) do exec(k-1, n, pred, program, [n]) |> Enum.reverse end defp exec(0, _, _, _, steps), do: steps defp exec(k, n, pred, program, steps) do case step(n, program) do :halt -> steps m -> if pred.(m), do: exec(k-1, m, pred, program, [m|steps]), else: exec(k, m, pred, program, steps) end end def is_pow2(n), do: band(n, n-1) == 0 def lowbit(n), do: lowbit(n, 0) defp lowbit(n, k) do case band(n, 1) do 0 -> lowbit(bsr(n, 1), k + 1) 1 -> k end end # rational multiplication defp mulrat({a, b}, {c, d}) do {p, q} = {a*c, b*d} g = gcd(p, q) {div(p, g), div(q, g)} end defp gcd(a, 0), do: a defp gcd(a, b), do: gcd(b, rem(a, b)) end primegen = Fractran.load("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") IO.puts "The first few states of the Fractran prime automaton are:\n#{inspect Fractran.exec(20, 2, primegen)}\n" prime = Fractran.exec(26, 2, &Fractran.is_pow2/1, primegen) |> Enum.map(&Fractran.lowbit/1) |> tl IO.puts "The first few primes are:\n#{inspect prime}"  Output: The first few states of the Fractran prime automaton are: [2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4] The first few primes are: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]  ## Erlang The exec() function can be passed a predicate which filters steps that satisfy a condition, which for the prime automata is a check to see if the number is a power of 2. #! /usr/bin/escript -mode(native). -import(lists, [map/2, reverse/1]). binary_to_ratio(B) -> {match, [_, Num, Den]} = re:run(B, "([0-9]+)/([0-9]+)"), {binary_to_integer(binary:part(B, Num)), binary_to_integer(binary:part(B, Den))}. load(Program) -> map(fun binary_to_ratio/1, re:split(Program, "[ ]+")). step(_, []) -> halt; step(N, [F|Fs]) -> {P, Q} = mulrat(F, {N, 1}), case Q of 1 -> P; _ -> step(N, Fs) end. exec(K, N, Program) -> reverse(exec(K - 1, N, fun (_) -> true end, Program, [N])). exec(K, N, Pred, Program) -> reverse(exec(K - 1, N, Pred, Program, [N])). exec(0, _, _, _, Steps) -> Steps; exec(K, N, Pred, Program, Steps) -> case step(N, Program) of halt -> Steps; M -> case Pred(M) of true -> exec(K - 1, M, Pred, Program, [M|Steps]); false -> exec(K, M, Pred, Program, Steps) end end. is_pow2(N) -> N band (N - 1) =:= 0. lowbit(N) -> lowbit(N, 0). lowbit(N, K) -> case N band 1 of 0 -> lowbit(N bsr 1, K + 1); 1 -> K end. main(_) -> PrimeGen = load("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"), io:format("The first few states of the Fractran prime automaton are: ~p~n~n", [exec(20, 2, PrimeGen)]), io:format("The first few primes are: ~p~n", [tl(map(fun lowbit/1, exec(26, 2, fun is_pow2/1, PrimeGen)))]). % rational multiplication mulrat({A, B}, {C, D}) -> {P, Q} = {A*C, B*D}, G = gcd(P, Q), {P div G, Q div G}. gcd(A, 0) -> A; gcd(A, B) -> gcd(B, A rem B).  Output: └─$ ▶ ./fractran.erl
The first few states of the Fractran prime automaton are: [2,15,825,725,1925,
2275,425,390,330,
290,770,910,170,
156,132,116,308,
364,68,4]

The first few primes are: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,
67,71,73,79,83,89,97]



## Factor

USING: io kernel math math.functions math.parser multiline
prettyprint sequences splitting ;
IN: rosetta-code.fractran

STRING: fractran-string
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
;

: fractran-parse ( str -- seq )
" \n" split [ string>number ] map ;

: fractran-step ( seq n -- seq n'/f )
2dup [ * integer? ] curry find nip dup [ * ] [ nip ] if ;

: fractran-run-full ( seq n -- )
[ dup ] [ dup . fractran-step ] while 2drop ;

: fractran-run-limited ( seq n steps -- )
[ dup pprint bl fractran-step ] times 2drop nl ;

: fractran-primes ( #primes seq n -- )
[ pick zero? ] [
dup 2 logn dup [ floor 1e-9 ~ ] [ 1. = not ] bi and [
dup 2 logn >integer pprint bl [ 1 - ] 2dip
] when fractran-step
] until 3drop nl ;

: main ( -- )
fractran-string fractran-parse 2
[ "First 20 numbers: " print 20 fractran-run-limited nl ]
[ "First 20 primes: " print [ 20 ] 2dip fractran-primes ]
2bi ;

MAIN: main

Output:
First 20 numbers:
2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4

First 20 primes:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71


## Fermat

Func FT( arr, n, m ) =
;{executes John H. Conway's FRACTRAN language for a program stored in [arr], an}
;{input integer stored in n, for a maximum of m steps}
;{To allow the program to run indefinitely, give it negative or noninteger m}
exec:=1;                 {boolean to track whether the program needs to halt}
len:=Cols[arr];          {length of the input program}
while exec=1 and m<>0 do
m:-;
!!n;                 {output the memory}
i:=1;                {index variable}
exec:=0;
while i<=len and exec=0 do
nf:=n*arr[i];
if Denom(nf) = 1 then
n:=nf;       {did we find an instruction to execute?}
exec:=1
fi;
i:+;
od;
od;
.;

;{Here is the program to run}
[arr]:=[( 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 )];

FT( [arr], 2, 20 );
Output:

2
15
825
725
1925
2275
425
390
330
290
770
910
170
156
132
116
308
364
68
4



## Fortran

### The Plan

As ever, how long is a piece of string: what might be the upper limit for the list of fractions? Similarly, obtaining the source code to be interpreted is a nuisance since as ever, one doesn't know how long it might be and performing a scan to find out will require more messing about than the activity of interest. One could compile in the specified example, but this is not very flexible even though a re-compilation will be swift. So, a pre-emptive requirement: first specify the number of fractions to be read in. Then reading the fractions has its annoyances also (aside from copying from the example text carrying along all the .html formatting) since given 2/3 a slash is not a suitable delimiter for free-format input, indeed it signifies end-of-input. Again, one can devise an input procedure involving a suitable scan, with more code drowning the stuff specific to the task. So, replace the slashes with a space, an acceptable free-format delimiter. As a comma is also acceptable, use those between the fractions, so, 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 Then, all that remains is to specify the starting number, and as well, a step limit. Naturally, the output can be in a more gracious form. It is presented as the input is read so that should something go awry there would be some indication of what was going on.

### The Code

The source style is F77 except for the use of the I0 format code, though not all F77 compilers will offer INTEGER*8. By not using the MODULE scheme, array parameters can't be declared via P(:) which implies a secret additional parameter giving the size of the array and which can be accessed via the likes of UBOUND(P, DIM = 1) Instead, the old-style specification involves no additional parameters and can be given as P(*) meaning "no statement" as to the upper bound, or P(M) which may be interpreted as the upper bound being the value of M in the compilers that allow this. The actual upper bound of the parameter is unknown and unchecked, so the older style of P(12345) or similar might be used. Rather to my surprise, this compiler (Compaq F90/95) complained if parameter M was declared after the arrays P(M),Q(M) as it is my habit to declare parameters in the order of their appearance.

C:\Nicky\RosettaCode\FRACTRAN\FRACTRAN.for(6) : Warning: This name has not been given an explicit type.   [M]
INTEGER P(M),Q(M)!The terms of the fractions.


So much for multi-pass compilers!

Similarly, without the MODULE protocol, in all calling routines function FRACTRAN would be deemed floating-point so a type declaration is needed in each.

      INTEGER FUNCTION FRACTRAN(N,P,Q,M)	!Notion devised by J. H. Conway.
Careful: the rule is N*P/Q being integer. N*6/3 is integer always because this is N*2/1, but 3 may not divide N.
Could check GCD(P,Q), dividing out the common denominator so MOD(N,Q) works.
INTEGER*8 N	!The work variable. Modified!
INTEGER M	!The number of fractions supplied.
INTEGER P(M),Q(M)!The terms of the fractions.
INTEGER I	!A stepper.
DO I = 1,M	!Search the supplied fractions, P(i)/Q(i).
IF (MOD(N,Q(I)).EQ.0) THEN	!Does the denominator divide N?
N = N/Q(I)*P(I)	!Yes, compute N*P/Q but trying to dodge overflow.
FRACTRAN = I	!Report the hit.
RETURN		!Done!
END IF	!Otherwise,
END DO		!Try the next fraction in the order supplied.
FRACTRAN = 0	!No hit.
END FUNCTION FRACTRAN	!That's it! Even so, "Turing complete"...

PROGRAM POKE
INTEGER FRACTRAN		!Not the default type of function.
INTEGER P(66),Q(66)	!Holds the fractions as P(i)/Q(i).
INTEGER*8 N		!The working number.
INTEGER I,IT,L,M		!Assistants.

WRITE (6,1)	!Announce.
1 FORMAT ("Interpreter for J.H. Conway's FRACTRAN language.")

Chew into an example programme.
OPEN (10,FILE = "Fractran.txt",STATUS="OLD",ACTION="READ")	!Rather than compiled-in stuff.
READ (10,*) L	!I need to know this without having to scan the input.
WRITE (6,2) L	!Reveal in case of trouble.
2 FORMAT (I0," fractions, as follow:")	!Should the input evoke problems.
READ (10,*) (P(I),Q(I),I = 1,L)	!Ask for the specified number of P,Q pairs.
WRITE (6,3) (P(I),Q(I),I = 1,L)	!Show what turned up.
3 FORMAT (24(I0,"/",I0:", "))	!As P(i)/Q(i) pairs. The colon means that there will be no trailing comma.
READ (10,*) N,M			!The start value, and the step limit.
CLOSE (10)			!Finished with input.
WRITE (6,4) N,M			!Hopefully, all went well.

Commence.
WRITE (6,10) 0,N		!Splat a heading.
10 FORMAT (/,"  Step  #F: N",/,I6,4X,": ",I0)	!Matched FORMAT 11.
DO I = 1,M		!Here we go!
IT = FRACTRAN(N,P,Q,L)		!Do it!
WRITE (6,11) I,IT,N		!Show it!
11   FORMAT (I6,I4,": ",I0)		!N last, as it may be big.
IF (IT.LE.0) EXIT		!No hit, so quit.
END DO			!The next step.
END	!Whee!


### The Results

Output:

Interpreter for J.H. Conway's FRACTRAN language.
14 fractions, as follow:
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

Step  #F: N
0    : 2
1  13: 15
2  14: 825
3   5: 725
4   6: 1925
5  11: 2275
6   1: 425
7   2: 390
8  10: 330
9   5: 290
10   6: 770
11  11: 910
12   1: 170
13   2: 156
14  10: 132
15   5: 116
16   6: 308
17  11: 364
18   1: 68
19   9: 4
20  13: 30
21  13: 225
22  14: 12375
23   5: 10875
24   6: 28875
25   5: 25375
26   6: 67375
27  11: 79625
28   1: 14875


Later Fortrans might offer the library function POPCNT(n) which returns the number of on-bits in an integer, most convenient for detecting a straight power of two in a binary computer. Adjusting the interpretation loop to be

      DO I = 1,M		!Here we go!
IT = FRACTRAN(N,P,Q,L)		!Do it!
IF (POPCNT(N).EQ.1) WRITE (6,11) I,IT,N		!Show it!
11   FORMAT (I6,I4,": ",I0)		!N last, as it may be big.
IF (IT.LE.0) EXIT		!No hit, so quit.
IF (N.LE.0) THEN		!Otherwise, worry about overflow.
WRITE (6,*) "Integer overflow!"	!Justified. The test is not certain.
WRITE (6,11) I,IT,N			!Alas, the step failed.
EXIT					!Give in.
END IF				!So much for overflow.
END DO			!The next step.


Interpreter for J.H. Conway's FRACTRAN language.
14 fractions, as follow:
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

Step  #F: N
0    : 2
19   9: 4
69   9: 8
280   9: 32
707   9: 128
2363   9: 2048
Integer overflow!
2863   6: -3816591145242754741


And no, fraction #9 is not always the one with the power of two only, for instance step 130 with N = 24x7. So, even a 64-bit integer is not enough, and with a 32-bit integer,

 Integer overflow!
296   6: -1406264171


In the absence of a proper test such as the IF OVERFLOW ...  of First Fortran (1957), the check for overflow relies on the improper appearance of negative numbers during two's-complement binary arithmetic that should only produce positive numbers. This will catch about half the events: N*M might or might not leave the sign bit on when the result overflows.

### Revised Plan

One could introduce a scheme for multi-precision (or "bignum") arithmetic, but there is no need to maintain a unified number N because there are no additions or subtractions, only multiplications and division without remainders. In other words, why not mess about with collections of prime factors? That is, represent N, P, and Q as the list of prime numbers with their powers for each case. To calculate N*P, just add the list of prime number powers for P to that for N, and similarly, for N/Q subtract. At no stage (except perhaps to decorate output or for debugging) need the list be multiplied out to give the unified number and so there is no need for multi-precision multiplies and divides. To determine if N is divisible by Q (that is, if N*fraction = N*P/Q is integral), check only that the primes listed by Q have powers no greater than those of the corresponding primes for N.

To facilitate access without searching the list of primes for N, instead its list is represented by an array of powers, NPPOW, with most entries zero. Thus, NPPOW(i) has the power for PRIME(i) in the factorisation of N, and for N = 2, NPPOW(1) = 1 with all other elements zero. But for FP (the factorisation of P) and FQ (for Q) there is a proper list provided via type FACTORED, whereby FP.PLIST(0) is the count of prime factors and FP.PLIST(1:FP.PLIST(0)) fingers the prime numbers that are factors, with FP.PPOW(i) having their corresponding powers. Thus, FP.PLIST(2) has the index of the second prime factor of P (should it have so many), which is PRIME(FP.PLIST(2)), and its power is FP.PPOW(2). Accordingly, to determine if N (as NPPOW) is divisible by one of the fractions FQ, the appropriate elements of NPPOW (that give its powers) must be compared to the corresponding powers in FQ.PPOW, and if ALL of the powers in NPPOW fingered by FQ.PLIST are greater than or equal to those in FQ.PPOW, then a hit! For this purpose, the factorisation of a prime number includes that prime number. Note that the prime number factorisation of one has no elements in its list, and happily, the ALL operation applied for no tests yields true as desired, because one divides any number. Thus, in the FRACTRAN op-code system, a fraction P/1 is always a match, and no fractions beyond it will ever be considered.

As a part of the preparation, for each fraction the greatest common divisor is divided out to remove the possibility that converting the test N times fraction being integral via fraction = P/Q to Q divides N will behave differently. For example, N*6/3 will always be integral, but N may not be divisible by three. Reducing 6/3 to 2/1 however will work as will reducing 55/25 to 11/5. The example contains no such occasions, but the possibility nags.

For output, the value of N will not be shown multiplied out but via a schedule showing the powers of the first few prime numbers that form its factorisation. Rather than staring in puzzlement at opaque monster strings of digits, one can view each separate prime factor's power counting up and down as the calculation proceeds. A simple scan of all the factorisations soon determines the highest prime employed, and this never changes. An extension of this checks for which primes are omitted, and in this example, none are. However, a further extension modifies the output of the schedule of powers so as to blank out those that are zero. This could be achieved by replacing the likes of WRITE (...) NPPOW by WRITE (...) I6FMT(NPPOW) where function I6FMT writes out its integer with I6 format if the value is positive, and supplies six spaces if not, were it not that few Fortran systems allow such re-entrant usage of the system for formatted output. So, prepare the output with a straightforward WRITE to a CHARACTER variable, blank out the portions where unwanted values appear, and write the result.

### Revised Code

Because this scheme requires a supply of prime numbers, it is convenient to employ the routines prepared for the extensible prime generator via module PRIMEBAG. So, this means escalating to the F90 style, and given that, some compound data structures can be used (for better mnemonics) in place of collections of arrays.

       MODULE CONWAYSIDEA	!Notion devised by J. H. Conway.
USE PRIMEBAG		!This is a common need.
INTEGER LASTP,ENUFF	!Some size allowances.
PARAMETER (LASTP = 66, ENUFF = 66)	!Should suffice for the example in mind.
INTEGER NPPOW(1:LASTP)	!Represent N as a collection of powers of prime numbers.
TYPE FACTORED		!But represent P and Q of freaction = P/Q
INTEGER PNUM(0:LASTP)	!As a list of prime number indices with PNUM(0) the count.
INTEGER PPOW(LASTP)	!And the powers. for the fingered primes.
END TYPE FACTORED	!Rather than as a simple number multiplied out.
TYPE(FACTORED) FP(ENUFF),FQ(ENUFF)	!Thus represent a factored fraction, P(i)/Q(i).
INTEGER PLIVE(ENUFF),NL	!Helps subroutine SHOWN display NPPOW.
CONTAINS		!Now for the details.
SUBROUTINE SHOWFACTORS(N)	!First, to show an internal data structure.
TYPE(FACTORED) N	!It is supplied as a list of prime factors.
INTEGER I		!A stepper.
DO I = 1,N.PNUM(0)	!Step along the list.
IF (I.GT.1) WRITE (MSG,"('x',$)") !Append a glyph for "multiply". WRITE (MSG,"(I0,$)") PRIME(N.PNUM(I))	!The prime fingered in the list.
IF (N.PPOW(I).GT.1) WRITE (MSG,"('^',I0,$)") N.PPOW(I) !With an interesting power? END DO !On to the next element in the list. WRITE (MSG,1) N.PNUM(0) !End the line 1 FORMAT (": Factor count ",I0) !With a count of prime factors. END SUBROUTINE SHOWFACTORS !Hopefully, this will not be needed often. TYPE(FACTORED) FUNCTION FACTOR(IT) !Into a list of primes and their powers. INTEGER IT,N !The number and a copy to damage. INTEGER P,POW !A stepper and a power. INTEGER F,NF !A factor and a counter. IF (IT.LE.0) STOP "Factor only positive numbers!" !Or else... N = IT !A copy I can damage. NF = 0 !No factors found. P = 0 !Because no primes have been tried. PP:DO WHILE (N.GT.1) !Step through the possibilities. P = P + 1 !Another prime impends. F = PRIME(P) !Grab a possible factor. POW = 0 !It has no power yet. FP:DO WHILE(MOD(N,F).EQ.0) !Well? POW = POW + 1 !Count a factor.. N = N/F !Reduce the number. END DO FP !The P'th prime's power's produced. IF (POW.GT.0) THEN !So, was it a factor? IF (NF.GE.LASTP) THEN !Yes. Have I room in the list? WRITE (MSG,1) IT,LASTP !Alas. 1 FORMAT ("Factoring ",I0," but with provision for only ", 1 I0," prime factors!") FACTOR.PNUM(0) = NF !Place the count so far, CALL SHOWFACTORS(FACTOR)!So this can be invoked. STOP "Not enough storage!" !Quite. END IF !But normally, NF = NF + 1 !Admit another factor. FACTOR.PNUM(NF) = P !Identify the prime. NOT the prime itself. FACTOR.PPOW(NF) = POW !Place its power. END IF !So much for that factor. END DO PP !Try another prime, if N > 1 still. FACTOR.PNUM(0) = NF !Place the count. END FUNCTION FACTOR !Thus, a list of primes and their powers. INTEGER FUNCTION GCD(I,J) !Greatest common divisor. INTEGER I,J !Of these two integers. INTEGER N,M,R !Workers. N = MAX(I,J) !Since I don't want to damage I or J, M = MIN(I,J) !These copies might as well be the right way around. 1 R = MOD(N,M) !Divide N by M to get the remainder R. IF (R.GT.0) THEN !Remainder zero? N = M !No. Descend a level. M = R !M-multiplicity has been removed from N. IF (R .GT. 1) GO TO 1 !No point dividing by one. END IF !If R = 0, M divides N. GCD = M !There we are. END FUNCTION GCD !Euclid lives on! INTEGER FUNCTION FRACTRAN(L) !Applies Conway's idea to a list of fractions. Could abandon all parameters since global variables have the details... INTEGER L !The last fraction to consider. INTEGER I,NF !Assistants. DO I = 1,L !Step through the fractions in the order they were given. NF = FQ(I).PNUM(0) !How many factors are listed in FQ(I)? IF (ALL(NPPOW(FQ(I).PNUM(1:NF)) !Can N (as NPPOW) be divided by Q (as FQ)? 1 .GE. FQ(I).PPOW(1:NF))) THEN !By comparing the supplies of prime factors. FRACTRAN = I !Yes! NPPOW(FQ(I).PNUM(1:NF)) = NPPOW(FQ(I).PNUM(1:NF)) !Remove prime powers from N 1 - FQ(I).PPOW(1:NF) !Corresponding to Q. NF = FP(I).PNUM(0) !Add powers to N NPPOW(FP(I).PNUM(1:NF)) = NPPOW(FP(I).PNUM(1:NF)) !Corresponding to P. 1 + FP(I).PPOW(1:NF) !Thus, N = N/Q*P. RETURN !That's all it takes! No multiplies nor divides! END IF !So much for that fraction. END DO !This relies on ALL(zero tests) yielding true, as when Q = 1. FRACTRAN = 0 !No hit. END FUNCTION FRACTRAN !No massive multi-precision arithmetic! SUBROUTINE SHOWN(S,F) !Service routine to show the state after a step is calculated. Could imaging a function I6FMT(23) that returns " 23" and " " for non-positive numbers. Can't do it, as if this were invoked via a WRITE statement, re-entrant use of WRITE usually fails. INTEGER S,F !Step number, Fraction number. INTEGER I !A stepper. CHARACTER*(9+4+1 + NL*6) ALINE !A scratchpad matching FORMAT 103. WRITE (ALINE,103) S,F,NPPOW(PLIVE(1:NL)) !Show it! 103 FORMAT (I9,I4,":",<NL>I6) !As a sequence of powers of primes. IF (F.LE.0) ALINE(10:13) = "" !Scrub when no fraction is fingered. DO I = 1,NL !Step along the live primes. IF (NPPOW(PLIVE(I)).GT.0) CYCLE !Ignoring the empowered ones. ALINE(15 + (I - 1)*6:14 + I*6) = "" !Blank out zero powers. END DO !On to the next. WRITE (MSG,"(A)") ALINE !Reveal at last. END SUBROUTINE SHOWN !A struggle. END MODULE CONWAYSIDEA !Simple... PROGRAM POKE USE CONWAYSIDEA !But, where does he get his ideas from? INTEGER P(ENUFF),Q(ENUFF) !Holds the fractions as P(i)/Q(i). INTEGER N !The working number. INTEGER LF !Last fraction given. INTEGER LP !Last prime needed. INTEGER MS !Maximum number of steps. INTEGER I,IT !Assistants. LOGICAL*1 PUSED(ENUFF) !Track the usage of prime numbers, MSG = 6 !Standard output. WRITE (6,1) !Announce. 1 FORMAT ("Interpreter for J. H. Conway's FRACTRAN language.") Chew into an example programme. 10 OPEN (10,FILE = "Fractran.txt",STATUS="OLD",ACTION="READ") !Rather than compiled-in stuff. READ (10,*) LF !I need to know this without having to scan the input. WRITE (MSG,11) LF !Reveal in case of trouble. 11 FORMAT (I0," fractions, as follow:") !Should the input evoke problems. READ (10,*) (P(I),Q(I),I = 1,LF) !Ask for the specified number of P,Q pairs. WRITE (MSG,12) (P(I),Q(I),I = 1,LF) !Show what turned up. 12 FORMAT (24(I0,"/",I0:", ")) !As P(i)/Q(i) pairs. The colon means that there will be no trailing comma. READ (10,*) N,MS !The start value, and the step limit. CLOSE (10) !Finished with input. WRITE (MSG,13) N,MS !Hopefully, all went well. 13 FORMAT ("Start with N = ",I0,", step limit ",I0) IF (.NOT.GRASPPRIMEBAG(66)) STOP "Gan't grab my file of primes!" !Attempt in hope. Convert the starting number to a more convenient form, an array of powers of successive prime numbers. 20 FP(1) = FACTOR(N) !Borrow one of the factor list variables. NPPOW = 0 !Clear all prime factor counts. DO I = 1,FP(1).PNUM(0) !Now find what they are. NPPOW(FP(1).PNUM(I)) = FP(1).PPOW(I) !Convert from a variable-length list END DO !To a fixed-length random-access array. PUSED = NPPOW.GT.0 !Note which primes have been used. LP = FP(1).PNUM(FP(1).PNUM(0)) !Recall the last prime required. More later. Convert the supplied P(i)/Q(i) fractions to lists of prime number factors and powers in FP(i) and FQ(i). DO I = 1,LF !Step through the fractions. IT = GCD(P(I),Q(I)) !Suspicion. IF (IT.GT.1) THEN !Justified? WRITE (MSG,21) I,P(I),Q(I),IT !Alas. Complain. The rule is N*(P/Q) being integer. 21 FORMAT ("Fraction ",I3,", ",I0,"/",I0,!N*6/3 is integer always because this is N*2/1, but 3 may not divide N. 1 " has common factor ",I0,"!") !By removing IT, P(I) = P(I)/IT !The test need merely check if N is divisible by Q. Q(I) = Q(I)/IT !And, as N is factorised in NPPOW END IF !And Q in FQ, subtractions of powers only is needed. FP(I) = FACTOR(P(I)) !Righto, form the factor list for P. PUSED(FP(I).PNUM(1:FP(I).PNUM(0))) = .TRUE. !Mark which primes it fingers. LP = MAX(LP,FP(I).PNUM(FP(I).PNUM(0))) !One has no prime factors: PNUM(0) = 0. FQ(I) = FACTOR(Q(I)) !And likewise for Q. PUSED(FQ(I).PNUM(1:FQ(I).PNUM(0))) = .TRUE. !Some primes may be omitted. LP = MAX(LP,FQ(I).PNUM(FQ(I).PNUM(0))) !If no prime factors, PNUM(0) fingers element zero, which is zero. END DO !All this messing about saves on multiplication and division. Check which primes are in use, preparing an index of live primes.. NL = 0 !No live primes. DO I = 1,LP !Check up to the last prime. IF (PUSED(I)) THEN !This one used? NL = NL + 1 !Yes. Another. PLIVE(NL) = I !Fingered. END IF !So much for that prime. END DO !On to the next. WRITE (MSG,22) NL,LP,PRIME(LP) !Remark on usage. 22 FORMAT ("Require ",I0," primes only, up to Prime(",I0,") = ",I0) !Presume always more than one prime. IF (LP.GT.LASTP) STOP "But, that's too many for array NPPOW!" Cast forth a heading. 100 WRITE (MSG,101) (PRIME(PLIVE(I)), I = 1,NL) !Splat a heading. 101 FORMAT (/,14X,"N as powers of prime factors",/, !The prime heading, 1 5X,"Step F#:",<LP>I6) !With primes beneath. CALL SHOWN(0,0) !Initial state of N as NPPOW. Step zero, no fraction. Commence! DO I = 1,MS !Here we go! IT = FRACTRAN(LF) !Do it! CALL SHOWN(I,IT) !Show it! IF (IT.LE.0) EXIT !Quit it? END DO !The next step. Complete! END !Whee!  ### Revised Results Edited in are >>> markers for the prime powers of two. Interpreter for J. H. Conway's FRACTRAN language. 14 fractions, as follow: 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 Start with N = 2, step limit 100 Require 10 primes only, up to Prime(10) = 29 N as powers of prime factors Step F#: 2 3 5 7 11 13 17 19 23 29 0 : 1 1 13: 1 1 2 14: 1 2 1 3 5: 2 1 4 6: 2 1 1 5 11: 2 1 1 6 1: 2 1 7 2: 1 1 1 1 8 10: 1 1 1 1 9 5: 1 1 1 10 6: 1 1 1 1 11 11: 1 1 1 1 12 1: 1 1 1 13 2: 2 1 1 14 10: 2 1 1 15 5: 2 1 16 6: 2 1 1 17 11: 2 1 1 18 1: 2 1 >>> 19 9: 2 20 13: 1 1 1 21 13: 2 2 22 14: 2 3 1 23 5: 1 3 1 24 6: 1 3 1 1 25 5: 3 1 1 26 6: 3 2 1 27 11: 3 2 1 28 1: 3 1 1 29 2: 1 1 2 1 1 30 1: 1 1 2 1 31 2: 2 2 1 1 32 10: 2 2 1 1 33 5: 2 1 1 1 34 6: 2 1 1 1 1 35 5: 2 1 1 1 36 6: 2 1 2 1 37 11: 2 1 2 1 38 1: 2 1 1 1 39 2: 3 1 1 1 40 1: 3 1 1 41 3: 3 1 42 4: 2 1 43 7: 2 1 1 44 4: 1 1 1 45 7: 1 2 1 46 4: 2 1 47 7: 3 1 48 8: 3 1 1 49 11: 3 1 1 50 1: 3 1 51 2: 1 1 2 1 52 10: 1 1 2 1 53 5: 1 2 1 54 6: 1 2 1 1 55 11: 1 2 1 1 56 1: 1 2 1 57 2: 2 1 1 1 58 10: 2 1 1 1 59 5: 2 1 1 60 6: 2 1 1 1 61 11: 2 1 1 1 62 1: 2 1 1 63 2: 3 1 1 64 10: 3 1 1 65 5: 3 1 66 6: 3 1 1 67 11: 3 1 1 68 1: 3 1 >>> 69 9: 3 70 13: 2 1 1 71 13: 1 2 2 72 13: 3 3 73 14: 3 4 1 74 5: 2 4 1 75 6: 2 4 1 1 76 5: 1 4 1 1 77 6: 1 4 2 1 78 5: 4 2 1 79 6: 4 3 1 80 11: 4 3 1 81 1: 4 2 1 82 2: 1 1 3 2 1 83 1: 1 1 3 1 1 84 2: 2 2 2 1 1 85 1: 2 2 2 1 86 2: 3 3 1 1 87 10: 3 3 1 1 88 5: 3 2 1 1 89 6: 3 2 1 1 1 90 5: 3 1 1 1 1 91 6: 3 1 1 2 1 92 5: 3 1 2 1 93 6: 3 1 3 1 94 11: 3 1 3 1 95 1: 3 1 2 1 96 2: 4 1 2 1 97 1: 4 1 1 1 98 3: 4 1 1 99 4: 3 1 1 100 7: 3 1 1 1  This time, restricting output to only occasions when N is a power of two requires no peculiar bit-counting function. Just change the interpretation loop to  DO I = 1,MS !Here we go! IT = FRACTRAN(LF) !Do it! IF (ALL(NPPOW(2:LP).EQ.0)) CALL SHOWN(I,IT) !Show it! IF (IT.LE.0) EXIT !Quit it? END DO !The next step.  Output: Interpreter for J. H. Conway's FRACTRAN language. 14 fractions, as follow: 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 Start with N = 2, step limit 6666666 Require 10 primes only, up to Prime(10) = 29 N as powers of prime factors Step F#: 2 3 5 7 11 13 17 19 23 29 0 : 1 19 9: 2 69 9: 3 280 9: 5 707 9: 7 2363 9: 11 3876 9: 13 8068 9: 17 11319 9: 19 19201 9: 23 36866 9: 29 45551 9: 31 75224 9: 37 101112 9: 41 117831 9: 43 152025 9: 47 215384 9: 53 293375 9: 59 327020 9: 61 428553 9: 67 507519 9: 71 555694 9: 73 700063 9: 79 808331 9: 83 989526 9: 89 1273490 9: 97 1434366 9: 101 1530213 9: 103 1710923 9: 107 1818254 9: 109 2019962 9: 113 2833089 9: 127 3104685 9: 131 3546320 9: 137 3720785 9: 139 4549718 9: 149 4755581 9: 151 5329874 9: 157 5958403 9: 163 6400897 9: 167 Execution took about two seconds. ### Add and Multiply Examples taken from Wikipaedia... Interpreter for J. H. Conway's FRACTRAN language. 1 fraction, as follows: 3/2 Start with N = 72, step limit 66 Require 2 primes only, up to Prime(2) = 3 N as powers of prime factors Step F#: 2 3 0 : 3 2 1 1: 2 3 2 1: 1 4 3 1: 5 4 : 5  The initial value is 72 = 23x32 so that "register" two holds 3 and register three holds 2. On completion, register three holds 5, the sum of 2 and 3. Interpreter for J. H. Conway's FRACTRAN language. 6 fractions, as follow: 455/33, 11/13, 1/11, 3/7, 11/2, 1/3 Start with N = 72, step limit 66 Require 6 primes only, up to Prime(6) = 13 N as powers of prime factors Step F#: 2 3 5 7 11 13 0 : 3 2 1 5: 2 2 1 2 1: 2 1 1 1 1 3 2: 2 1 1 1 1 4 1: 2 2 2 1 5 2: 2 2 2 1 6 3: 2 2 2 7 4: 2 1 2 1 8 4: 2 2 2 9 5: 1 2 2 1 10 1: 1 1 3 1 1 11 2: 1 1 3 1 1 12 1: 1 4 2 1 13 2: 1 4 2 1 14 3: 1 4 2 15 4: 1 1 4 1 16 4: 1 2 4 17 5: 2 4 1 18 1: 1 5 1 1 19 2: 1 5 1 1 20 1: 6 2 1 21 2: 6 2 1 22 3: 6 2 23 4: 1 6 1 24 4: 2 6 25 6: 1 6 26 6: 6 27 : 6  Here, register two holds 3 and register three holds 2. Their product appears in register five. ## FreeBASIC Added a compiler condition to make the program work with the old GMP.bi header file ' version 06-07-2015 ' compile with: fbc -s console ' uses gmp #Include Once "gmp.bi" ' in case the two #define's are missing from 'gmp.bi' define them now #Ifndef mpq_numref #Define mpq_numref(Q) (@(Q)->_mp_num) #Define mpq_denref(Q) (@(Q)->_mp_den) #EndIf Dim As String prog(0 To ...) = {"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"} Dim As UInteger i, j, c, max = UBound(prog) Dim As Integer scanbit Dim As ZString Ptr gmp_str : gmp_str = Allocate(10000) Dim As Mpq_ptr in_, out_ in_ = Allocate(Len(__mpq_struct)) : Mpq_init(in_) out_ = Allocate(Len(__mpq_struct)) : Mpq_init(out_) Dim As mpz_ptr num, den num = Allocate(Len(__mpz_struct)) : Mpz_init(num) den = Allocate(Len(__mpz_struct)) : Mpz_init(den) Dim As mpq_ptr instruction(max) For i = 0 To max instruction(i) = Allocate(Len(__mpq_struct)) mpq_init(instruction(i)) mpq_set_str(instruction(i), prog(i), 10 ) Next mpq_set_str(in_ ,"2",10) i = 0 : j = 0 Print "2"; Do mpq_mul(out_, instruction(i), in_) i = i + 1 den = mpq_denref(out_) If mpz_cmp_ui(den, 1) = 0 Then Mpq_get_str(gmp_str, 10, out_) Print ", ";*gmp_str; mpq_swap(in_, out_) i = 0 j = j + 1 End If Loop Until j > 14 ' this one only display if the integer is 2^p, p being prime mpq_set_str(in_ ,"2",10) i = 0 : j = 0 : c = 0 Print : Print : Print Print "count iterations prime 2^prime" Do mpq_mul(out_, instruction(i), in_) i = i + 1 j = j + 1 den = mpq_denref(out_) If mpz_cmp_ui(den, 1) = 0 Then num = mpq_numref(out_) scanbit = mpz_scan1(num, 0) ' if scanbit = 0 then number is odd If scanbit > 0 Then ' return from mpz_scan1(num, scanbit+1) is -1 for power of 2 If mpz_scan1(num, scanbit +1) = -1 Then If c <= 20 Then Mpq_get_str(gmp_str, 10, out_) Else *gmp_str = "" c = c + 1 Print Using "##### ################### ######## "; c; j; scanbit; Print *gmp_str If InKey <> "" Then Exit Do End If End If mpq_swap(in_, out_) i = 0 End If Loop ' Loop Until scanbit > 300 ' Loop Until InKey <> "" ' Loop Until scanbit > 300 Or InKey <> "" ' stopping conditions will slow down the hole loop ' loop will check for key if it's printing a result ' empty keyboard buffer While InKey <> "" : Wend Print : Print "hit any key to end program" Sleep End  Output: 2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116 count iterations prime 2^prime 1 129 2 4 2 425 3 8 3 1563 5 32 4 3735 7 128 5 11674 11 2048 6 18811 13 8192 7 38010 17 131072 8 52854 19 524288 9 88134 23 8388608 10 166070 29 536870912 11 204575 31 2147483648 12 333931 37 137438953472 13 446506 41 2199023255552 14 519556 43 8796093022208 15 667496 47 140737488355328 16 940183 53 9007199254740992 17 1274660 59 576460752303423488 18 1419935 61 2305843009213693952 19 1853979 67 147573952589676412928 20 2191673 71 2361183241434822606848 shorten output file 42 34533967 181 43 40326168 191 ## Fōrmulæ Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition. Programs in Fōrmulæ are created/edited online in its website. In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation. Solution It is a function that accepts the program to run (as a list), the initial value of n and the number of values to generate. It uses a local nested function next() that calculates the next value of . If it can be calculated, it is added to a result array and return true, elsewhere return false. The main work is to iterate while the next() returns true and the number of values to generate is not reached. The following is the call with the program for primes, initial n value of 2, and returning 20 values: Bonus using the previous FRACTAN program to generate the first 20 primes. It requires a modification to the previous program. FRACTRAN program for addition FRACTRAN program for multiplication ## Go Basic task: This compiles to produce a program that reads the limit, starting number n, and list of fractions as command line arguments, with the list of fractions as a single argument. package main import ( "fmt" "log" "math/big" "os" "strconv" "strings" ) func compile(src string) ([]big.Rat, bool) { s := strings.Fields(src) r := make([]big.Rat, len(s)) for i, s1 := range s { if _, ok := r[i].SetString(s1); !ok { return nil, false } } return r, true } func exec(p []big.Rat, n *big.Int, limit int) { var q, r big.Int rule: for i := 0; i < limit; i++ { fmt.Printf("%d ", n) for j := range p { q.QuoRem(n, p[j].Denom(), &r) if r.BitLen() == 0 { n.Mul(&q, p[j].Num()) continue rule } } break } fmt.Println() } func usage() { log.Fatal("usage: ft <limit> <n> <prog>") } func main() { if len(os.Args) != 4 { usage() } limit, err := strconv.Atoi(os.Args[1]) if err != nil { usage() } var n big.Int _, ok := n.SetString(os.Args[2], 10) if !ok { usage() } p, ok := compile(os.Args[3]) if !ok { usage() } exec(p, &n, limit) }  Command line usage, with program compiled as "ft": > ft 15 2 "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 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132  Extra credit: This invokes above program with appropriate arguments, and processes the output to obtain the 20 primes. package main import ( "fmt" "log" "math/big" "os" "os/exec" ) func main() { c := exec.Command("ft", "1000000", "2", 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) c.Stderr = os.Stderr r, err := c.StdoutPipe() if err != nil { log.Fatal(err) } if err = c.Start(); err != nil { log.Fatal(err) } var n big.Int for primes := 0; primes < 20; { if _, err = fmt.Fscan(r, &n); err != nil { log.Fatal(err) } l := n.BitLen() - 1 n.SetBit(&n, l, 0) if n.BitLen() == 0 && l > 1 { fmt.Printf("%d ", l) primes++ } } fmt.Println() }  Output: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71  ## Haskell ### Running the program import Data.List (find) import Data.Ratio (Ratio, (%), denominator) fractran :: (Integral a) => [Ratio a] -> a -> [a] fractran fracts n = n : case find (\f -> n mod denominator f == 0) fracts of Nothing -> [] Just f -> fractran fracts$ truncate (fromIntegral n * f)


Example:

λ> let prog = [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]

λ> take 15 $fractran prog 2 [2,15,825,725,1925,2275,425,390,330,290,770,910,170,156,132] ### Reading the program Additional import import Data.List.Split (splitOn)  readProgram :: String -> [Ratio Int] readProgram = map (toFrac . splitOn "/") . splitOn "," where toFrac [n,d] = read n % read d  Example of running the program: λ> let prog = readProgram "17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23 , 77/19, 1/17, 11/13, 13/11, 15/14, 15/2, 55/1" λ> take 15$ fractran prog 2
[2,15,825,725,1925,2275,425,390,330,290,770,910,170,156,132]

### Generation of primes

import Data.Maybe (mapMaybe)
import Data.List (elemIndex)

primes :: [Int]
primes = mapMaybe log2 $fractran prog 2 where prog = [ 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 ] log2 = fmap succ . elemIndex 2 . takeWhile even . iterate (div 2)  λ> take 20 primes [1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67] ## Icon and Unicon Works in both languages: record fract(n,d) procedure main(A) fractran("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) end procedure fractran(s, n, limit) execute(parse(s),n, limit) end procedure parse(s) f := [] s ? while not pos(0) do { tab(upto(' ')|0) ? put(f,fract(tab(upto('/')), (move(1),tab(0)))) move(1) } return f end procedure execute(f,d,limit) /limit := 15 every !limit do { if d := (d%f[i := !*f].d == 0, (writes(" ",d)/f[i].d)*f[i].n) then {} else break write() } write() end  Output: ->fractan 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 ->  ## J ### Hybrid version Solution: toFrac=: '/r' 0&".@charsub ] NB. read fractions from string fractran15=: ({~ (= <.) i. 1:)@(toFrac@[ * ]) ^:(<15) NB. return first 15 Fractran results  Example:  taskstr=: '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' taskstr fractran15 2 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132  ### Tacit version Solution This is a variation of the previous solution which it is not entirely tacit due to the use of the explicit standard library verb (function) charsub. The adverb (functional) fractran is defined as a fixed tacit adverb (that is, a stateless point-free functional), fractran=. (((({~ (1 i.~ (= <.)))@:* ::]^:)(]))(".@:('1234567890r ' {~ '1234567890/ '&i.)@:[))(:6)  The argument of fractran specifies a limit for the number of steps; if the limit is boxed the intermediate results are also included in the result. Example  '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' (<15) fractran 2 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132  Extra credit The prime numbers are produced via the adverb primes; its argument has the same specifications as the argument for the fractran adverb (which is used in its definition), primes=. ('fractan'f.) ((1 }. 2 ^. (#~ *./@:e.&2 0"1@:q:))@:) '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' (<555555) primes 2 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71  primes is also a stateless point-free functional,  primes ((((({~ (1 i.~ (= <.)))@:* ::]^:)(]))(".@:('1234567890r ' {~ '1234567890/ '&i.)@:[))(:6))((1 }. 2 ^. (#~ *./@:e.&2 0"1@:q:))@:)  Turing completeness of J's stateless point-free dialect When _ is the limit argument (i.e., when no limit is imposed) the run will halt according to the FRACTRAN programming language specifications (the run might also be forced to halt if a trivial changeless single cycle, induced by a useless 1/1 fraction, is detected). Thus, the FRACTRAN associated verb (function) is,  _ fractran ".@:('1234567890r ' {~ '1234567890/ '&i.)@:[ ({~ (1 i.~ (= <.)))@:* ::]^:_ ]  Actually, most of the code above is there to comply with the task's requirement of a "natural format." When J's format for fractions is used the FRACTRAN verb becomes, FRACTRAN=. ({~ (1 i.~ (= <.)))@:* ::]^:_  which is an indirect concise confirmation that J's fixed tacit dialect is Turing complete. In the following example, FRACTRAN calculates the product 4 * 6, the initial value 11664 = (2^4)*(3^6) holds 4 in the register associated with 2 and holds 6 in the register associated with 3; the result 59604644775390625 = 5^24 holds the product 24 = 4 * 6 in the register associated with 5,  455r33 11r13 1r11 3r7 11r2 1r3 FRACTRAN 11664 59604644775390625  ## Java import java.util.Vector; import java.util.regex.Matcher; import java.util.regex.Pattern; public class Fractran{ public static void main(String []args){ new Fractran("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); } final int limit = 15; Vector<Integer> num = new Vector<>(); Vector<Integer> den = new Vector<>(); public Fractran(String prog, Integer val){ compile(prog); dump(); exec(2); } void compile(String prog){ Pattern regexp = Pattern.compile("\\s*(\\d*)\\s*\\/\\s*(\\d*)\\s*(.*)"); Matcher matcher = regexp.matcher(prog); while(matcher.find()){ num.add(Integer.parseInt(matcher.group(1))); den.add(Integer.parseInt(matcher.group(2))); matcher = regexp.matcher(matcher.group(3)); } } void exec(Integer val){ int n = 0; while(val != null && n<limit){ System.out.println(n+": "+val); val = step(val); n++; } } Integer step(int val){ int i=0; while(i<den.size() && val%den.get(i) != 0) i++; if(i<den.size()) return num.get(i)*val/den.get(i); return null; } void dump(){ for(int i=0; i<den.size(); i++) System.out.print(num.get(i)+"/"+den.get(i)+" "); System.out.println(); } }  ## JavaScript ### Imperative // Parses the input string for the numerators and denominators function compile(prog, numArr, denArr) { let regex = /\s*(\d*)\s*\/\s*(\d*)\s*(.*)/m; let result; while (result = regex.exec(prog)) { numArr.push(result[1]); denArr.push(result[2]); prog = result[3]; } return [numArr, denArr]; } // Outputs the result of the compile stage function dump(numArr, denArr) { let output = ""; for (let i in numArr) { output += ${numArr[i]}/${denArr[i]} ; } return ${output}<br>;
}

// Step
function step(val, numArr, denArr) {
let i = 0;
while (i < denArr.length && val % denArr[i] != 0) i++;
return numArr[i] * val / denArr[i];
}

// Executes Fractran
function exec(val, i, limit, numArr, denArr) {
let output = "";
while (val && i < limit) {
output += ${i}:${val}<br>;
val = step(val, numArr, denArr);
i++;
}
return output;
}

// Main
// Outputs to DOM (clears and writes at the body tag)
let body = document.body;
let [num, den] = compile("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", [], []);
body.innerHTML = dump(num, den);
body.innerHTML += exec(2, 0, 15, num, den);


### Functional

Here is a functionally composed version, which also derives a few primes. I may have missed something, but this first draft suggests that we may need bigInt support (which JS lacks) to get as far as the sixth prime.

(() => {
'use strict';

// fractran :: [Ratio Int] -> Int -> Gen [Int]
const fractran = (xs, n) => {
function* go(n) {
const p = r => 0 === v % r.d;
let
v = n,
mb = find(p, xs);
yield v
while (!mb.Nothing) {
mb = bindMay(
find(p, xs),
r => (
v = truncate({
type: 'Ratio',
n: v * r.n,
d: r.d
}),
Just(v)
)
);
mb.Just && (yield v)
}
};
return go(n);
};

// readRatios :: String -> [Ratio]
splitOn(',', s)
);

// main :: IO()
const main = () => {

// strRatios :: String
const strRatios = 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;

showLog(
'First fifteen steps:',
take(15,
)
);

showLog(
'First five primes:',
take(5,
mapMaybeGen(
x => fmapMay(
succ,
elemIndex(
2,
takeWhileGen(
even,
iterate(n => div(n, 2), x)
)
)
),
)
)
);
};

// GENERIC ABSTRACTIONS ----------------------------

// Just :: a -> Maybe a
const Just = x => ({
type: 'Maybe',
Nothing: false,
Just: x
});

// Nothing :: Maybe a
const Nothing = () => ({
type: 'Maybe',
Nothing: true,
});

// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});

// abs :: Num -> Num
const abs = Math.abs;

// bindMay (>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
const bindMay = (mb, mf) =>
mb.Nothing ? mb : mf(mb.Just);

// div :: Int -> Int -> Int
const div = (x, y) => Math.floor(x / y);

// elemIndex :: Eq a => a -> [a] -> Maybe Int
const elemIndex = (x, xs) => {
const i = xs.indexOf(x);
return -1 === i ? (
Nothing()
) : Just(i);
};

// even :: Int -> Bool
const even = n => 0 === n % 2;

// find :: (a -> Bool) -> [a] -> Maybe a
const find = (p, xs) => {
for (let i = 0, lng = xs.length; i < lng; i++) {
if (p(xs[i])) return Just(xs[i]);
}
return Nothing();
};

// fmapMay (<$>) :: (a -> b) -> Maybe a -> Maybe b const fmapMay = (f, mb) => mb.Nothing ? ( mb ) : Just(f(mb.Just)); // foldl :: (a -> b -> a) -> a -> [b] -> a const foldl = (f, a, xs) => xs.reduce(f, a); // gcd :: Int -> Int -> Int const gcd = (x, y) => { const _gcd = (a, b) => (0 === b ? a : _gcd(b, a % b)), abs = Math.abs; return _gcd(abs(x), abs(y)); }; // iterate :: (a -> a) -> a -> Gen [a] function* iterate(f, x) { let v = x; while (true) { yield(v); v = f(v); } } // map :: (a -> b) -> [a] -> [b] const map = (f, xs) => xs.map(f); // mapMaybeGen :: (a -> Maybe b) -> Gen [a] -> [b] function* mapMaybeGen(mf, gen) { let v = take(1, gen); while (0 < v.length) { let mb = mf(v[0]); if (!mb.Nothing) yield mb.Just v = take(1, gen); } } // properFracRatio :: Ratio -> (Int, Ratio) const properFracRatio = nd => { const [q, r] = Array.from(quotRem(nd.n, nd.d)); return Tuple(q, ratio(r, nd.d)); }; // quot :: Int -> Int -> Int const quot = (n, m) => Math.floor(n / m); // quotRem :: Int -> Int -> (Int, Int) const quotRem = (m, n) => Tuple(Math.floor(m / n), m % n); // ratio :: Int -> Int -> Ratio Int const ratio = (n, d) => 0 !== d ? (() => { const g = gcd(n, d); return { type: 'Ratio', 'n': quot(n, g), // numerator 'd': quot(d, g) // denominator } })() : undefined; // read :: Read a => String -> a const read = JSON.parse; // showLog :: a -> IO () const showLog = (...args) => console.log( args .map(JSON.stringify) .join(' -> ') ); // snd :: (a, b) -> b const snd = tpl => tpl[1]; // splitOn :: [a] -> [a] -> [[a]] // splitOn :: String -> String -> [String] const splitOn = (pat, src) => src.split(pat); // succ :: Int -> Int const succ = x => 1 + x; // take :: Int -> [a] -> [a] // take :: Int -> String -> String const take = (n, xs) => xs.constructor.constructor.name !== 'GeneratorFunction' ? ( xs.slice(0, n) ) : [].concat.apply([], Array.from({ length: n }, () => { const x = xs.next(); return x.done ? [] : [x.value]; })); // takeWhileGen :: (a -> Bool) -> Gen [a] -> [a] const takeWhileGen = (p, xs) => { const ys = []; let nxt = xs.next(), v = nxt.value; while (!nxt.done && p(v)) { ys.push(v); nxt = xs.next(); v = nxt.value } return ys; }; // truncate :: Num -> Int const truncate = x => 'Ratio' === x.type ? ( properFracRatio(x)[0] ) : properFraction(x)[0]; // MAIN --- return main(); })();  Output: "First fifteen steps:" -> [2,15,825,725,1925,2275,425,390,330,290,770,910,170,156,132] "First five primes:" -> [1,2,3,5,7] ## Julia Works with: Julia version 1.9 # FRACTRAN interpreter implemented as an iterable struct using .Iterators: filter, map, take struct Fractran rs::Vector{Rational{BigInt}} i₀::BigInt limit::Int end Base.iterate(f::Fractran, i = f.i₀) = for r in f.rs if iszero(i % r.den) i = i ÷ r.den * r.num return i, i end end interpret(f::Fractran) = take( map(trailing_zeros, filter(ispow2, f)) f.limit) Base.show(io::IO, f::Fractran) = join(io, interpret(f), ' ') macro code_str(s) [eval(Meta.parse(replace(t, "/" => "//"))) for t ∈ split(s)] end primes = Fractran(code"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, 30) # Output println("First 25 iterations of FRACTRAN program 'primes':\n2 ", join(take(primes, 25), ' ')) println("\nWatch the first 30 primes dropping out within seconds:") primes  Output: First 25 iterations of FRACTRAN program 'primes': 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4 30 225 12375 10875 28875 25375 Watch the first 30 primes dropping out within seconds: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113  ## Kotlin // version 1.1.3 import java.math.BigInteger class Fraction(val num: BigInteger, val denom: BigInteger) { operator fun times(n: BigInteger) = Fraction (n * num, denom) fun isIntegral() = num % denom == BigInteger.ZERO } fun String.toFraction(): Fraction { val split = this.split('/') return Fraction(BigInteger(split[0]), BigInteger(split[1])) } val BigInteger.isPowerOfTwo get() = this.and(this - BigInteger.ONE) == BigInteger.ZERO val log2 = Math.log(2.0) fun fractran(program: String, n: Int, limit: Int, primesOnly: Boolean): List<Int> { val fractions = program.split(' ').map { it.toFraction() } val results = mutableListOf<Int>() if (!primesOnly) results.add(n) var nn = BigInteger.valueOf(n.toLong()) while (results.size < limit) { val frac = fractions.find { (it * nn).isIntegral() } ?: break nn = nn * frac.num / frac.denom if (!primesOnly) { results.add(nn.toInt()) } else if (primesOnly && nn.isPowerOfTwo) { val prime = (Math.log(nn.toDouble()) / log2).toInt() results.add(prime) } } return results } fun main(args: Array<String>) { val program = "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" println("First twenty numbers:") println(fractran(program, 2, 20, false)) println("\nFirst twenty primes:") println(fractran(program, 2, 20, true)) }  Output: First twenty numbers: [2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4] First twenty primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]  ## Mathematica / Wolfram Language This isn't as efficient as possible for long lists of fractions, since it doesn't stop doing n*listelements once it finds an integer. Instead, it computes "is integer?" for n*{all list elements}. For short lists that's probably not a big deal. fractionlist = {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}; n = 2; steplimit = 20; j = 0; break = False; While[break == False && j <= steplimit, newlist = n fractionlist; isintegerlist = IntegerQ[#] & /@ newlist; truepositions = Position[isintegerlist, True]; If[Length[truepositions] == 0, break = True, Print[ToString[j] <> ": " <> ToString[n]]; n = newlist[[truepositions[[1, 1]]]]; j++; ] ]  Output: 0: 2 1: 15 2: 825 3: 725 4: 1925 5: 2275 6: 425 7: 390 8: 330 9: 290 10: 770 11: 910 12: 170 13: 156 14: 132 15: 116 16: 308 17: 364 18: 68 19: 4 20: 30 ### Functional Version Here is a different solution using a functional approach: fractran[ program : {__ ? (Element[#, PositiveRationals] &)}, (* list of positive fractions *) n0_Integer, (* initial state *) maxSteps : _Integer : Infinity] := (* max number of steps *) NestWhileList[ (* Return a list representing the evolution of the state n *) Function[n, SelectFirst[IntegerQ][program * n]], (* Select first integer in n*program, if none return Missing *) n0, Not @* MissingQ, (* continue while the state is not Missing *) 1, maxSteps]$PRIMEGAME = {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};
fractran[$PRIMEGAME, 2, 50]  Output: {2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4, 30, 225, 12375, 10875, 28875, 25375, 67375, 79625, 14875, 13650, 2550, 2340, 1980, 1740, 4620, 4060, 10780, 12740, 2380, 2184, 408, 152, 92, 380, 230, 950, 575, 2375, 9625, 11375, 2125} Extract the first 20 prime numbers encoded as powers of 2: Select[IntegerQ] @ Log2[fractran[$PRIMEGAME, 2, 500000]]

Output:
{1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67}

## Nim

### Using fractions

Library: bignum

This is a simple implementation which operates on fractions. As Nim standard library doesn’t provide a module for big numbers, we have used the extra library “bignum” which relies on “gmp”. We provide a general function to run any Fractran program and a specialized iterator to find prime numbers.

import strutils
import bignum

const PrimeProg = "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"

iterator values(prog: openArray[Rat]; init: Natural): Int =
## Run the program "prog" with initial value "init" and yield the values.
var n = newInt(init)
var next: Rat
while true:
for fraction in prog:
next = n * fraction
if next.denom == 1:
break
n = next.num
yield n

func toFractions(fractList: string): seq[Rat] =
## Convert a string to a list of fractions.
for f in fractList.split():

proc run(progStr: string; init: Natural; maxSteps: Natural = 0) =
## Run the program described by string "progStr" with initial value "init",
## stopping after "maxSteps" (0 means for ever).
## Display the value after each step.
let prog = progStr.toFractions()
var stepCount = 0
for val in prog.values(init):
inc stepCount
echo stepCount, ": ", val
if stepCount == maxSteps:
break

iterator primes(n: Natural): int =
# Yield the list of first "n" primes.
let prog = PrimeProg.toFractions()
var count = 0
for val in prog.values(2):
if isZero(val and (val - 1)):
# This is a power of two.
yield val.digits(2).int - 1   # Compute the exponent as number of binary digits minus one.
inc count
if count == n:
break

# Run the program to compute primes displaying values at each step and stopping after 10 steps.
echo "First ten steps for program to find primes:"
PrimeProg.run(2, 10)

# Find the first 20 primes.
echo "\nFirst twenty prime numbers:"
for val in primes(20):
echo val

Output:
First ten steps for program to find primes:
1: 15
2: 825
3: 725
4: 1925
5: 2275
6: 425
7: 390
8: 330
9: 290
10: 770

First twenty prime numbers:
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71


### Using decomposition in prime factors

With this algorithm, we no longer need big numbers. To avoid overflow, value at each step is displayed using its decomposition in prime factors.

import algorithm
import sequtils
import strutils
import tables

const PrimeProg = "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"

type
Fraction = tuple[num, denom: int]
Factors = Table[int, int]
FractranProg = object
primes: seq[int]
nums, denoms: seq[Factors]
exponents: seq[int]   # Could also use a CountTable.

iterator fractions(fractString: string): Fraction =
## Extract fractions from a string and yield them.
for f in fractString.split():
let fields = f.strip().split('/')
assert fields.len == 2
yield (fields[0].parseInt(), fields[1].parseInt())

iterator factors(val: int): tuple[val, exp: int] =
## Extract factors from a positive integer.

# Extract factor 2.
var val = val
var exp = 0
while (val and 1) == 0:
inc exp
val = val shr 1
if exp != 0:
yield (2, exp)

# Extract odd factors.
var d = 3
while d <= val:
exp = 0
while val mod d == 0:
inc exp
val = val div d
if exp != 0:
yield (d, exp)
inc d, 2

func newProg(fractString: string; init: int): FractranProg =
## Initialize a Fractran program.

for f in fractString.fractions():
# Extract numerators factors.
var facts: Factors
for (val, exp) in f.num.factors():
facts[val] = exp
# Extract denominator factors.
facts.clear()
for (val, exp) in f.denom.factors():
facts[val] = exp

# Finalize list of primes.
result.primes.sort()
result.primes = result.primes.deduplicate(true)

# Allocate and initialize exponent sequence.
result.exponents.setLen(result.primes[^1] + 1)
for (val, exp) in init.factors():
result.exponents[val] = exp

func doOneStep(prog: var FractranProg): bool =
## Execute one step of the program.

for idx, factor in prog.denoms:
block tryFraction:
for val, exp in factor.pairs():
if prog.exponents[val] < exp:
# Not a multiple of the denominator.
break tryFraction
# Divide by the denominator.
for val, exp in factor.pairs():
dec prog.exponents[val], exp
# Multiply by the numerator.
for val, exp in prog.nums[idx]:
inc prog.exponents[val], exp
return true

func $(prog: FractranProg): string = ## Display a value as a product of prime factors. for val, exp in prog.exponents: if exp != 0: if result.len > 0: result.add('.') result.add($val)
if exp > 1:
result.add($exp) proc run(fractString: string; init: int; maxSteps = 0) = ## Run a Fractran program. var prog = newProg(fractString, init) var stepCount = 0 while stepCount < maxSteps: if not prog.doOneStep(): echo "*** No more possible fraction. Program stopped." return inc stepCount echo stepCount, ": ", prog proc findPrimes(maxCount: int) = ## Search and print primes. var prog = newProg(PrimeProg, 2) let oddPrimes = prog.primes[1..^1] var primeCount = 0 while primeCount < maxCount: discard prog.doOneStep() block powerOf2: if prog.exponents[2] > 0: for p in oddPrimes: if prog.exponents[p] != 0: # Not a power of 2. break powerOf2 inc primeCount echo primeCount, ": ", prog.exponents[2] #------------------------------------------------------------------------------ echo "First ten steps for program to find primes:" run(PrimeProg, 2, 10) echo "\nFirst twenty prime numbers:" findPrimes(20)  Output: First ten steps for program to find primes: 1: 3.5 2: 3.5^2.11 3: 5^2.29 4: 5^2.7.11 5: 5^2.7.13 6: 5^2.17 7: 2.3.5.13 8: 2.3.5.11 9: 2.5.29 10: 2.5.7.11 First twenty prime numbers: 1: 2 2: 3 3: 5 4: 7 5: 11 6: 13 7: 17 8: 19 9: 23 10: 29 11: 31 12: 37 13: 41 14: 43 15: 47 16: 53 17: 59 18: 61 19: 67 20: 71  ## OCaml This reads a Fractran program from standard input (keyboard or file) and runs it with the input given by the command line arguments, using arbitrary-precision numbers and fractions. open Num let get_input () = num_of_int ( try int_of_string Sys.argv.(1) with _ -> 10) let get_max_steps () = try int_of_string Sys.argv.(2) with _ -> 50 let read_program () = let line = read_line () in let words = Str.split (Str.regexp " +") line in List.map num_of_string words let is_int n = n =/ (integer_num n) let run_program num prog = let replace n = let rec step = function | [] -> None | h :: t -> let n' = h */ n in if is_int n' then Some n' else step t in step prog in let rec repeat m lim = Printf.printf " %s\n" (string_of_num m); if lim = 0 then print_endline "Reached max step limit" else match replace m with | None -> print_endline "Finished" | Some x -> repeat x (lim-1) in let max_steps = get_max_steps () in repeat num max_steps let () = let num = get_input () in let prog = read_program () in run_program num prog  The program 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 saved in a file can can be run with the command: ocaml nums.cma str.cma fractran.ml 2 30 < primes.fct Output:  2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4 30 225 12375 10875 28875 25375 67375 79625 14875 13650 2550 Reached max step limit  ## PARI/GP In this version ideas were borrowed from C++, Java and JavaScript. Works with: PARI/GP version 2.7.4 and above \\ FRACTRAN \\ 4/27/16 aev fractran(val,ft,lim)={ my(ftn=#ft,fti,di,L=List(),j=0); while(val&&j<lim, listput(L,val); for(i=1,ftn, fti=ft[i]; di=denominator(fti); if(val%di==0, break));\\fend i val= numerator(fti)*val/di; j++);\\wend j return(Vec(L)); } {\\ Executing: my(v=[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]); print(fractran(2,v,15)); } Output: [2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132]  ## Perl Instead of printing all steps, I chose to only print those steps which were a power of two. This makes the fact that it's a prime-number-generating program much clearer. use strict; use warnings; use Math::BigRat; my ($n, @P) = map Math::BigRat->new($_), qw{ 2 17/91 78/85 19/51 23/38 29/33 77/29 95/23 77/19 1/17 11/13 13/11 15/14 15/2 55/1 };$|=1;
MAIN: for( 1 .. 5000 ) {
print " " if $_ > 1; my ($pow, $rest) = (0,$n->copy);
until( $rest->is_odd ) { ++$pow;
$rest->bdiv(2); } if($rest->is_one ) {
print "2**$pow"; } else { #print$n;
}
for my $f_i (@P) { my$nf_i = $n *$f_i;
next unless $nf_i->is_int;$n = $nf_i; next MAIN; } last; } print "\n";  If you uncomment the #print$n

, it will print all the steps.

## Phix

Using the same ideas as the Fortran entry (thanks!).

For example, suppose that known_factors happens to be {2,3,5}. [the exact order may vary]
2 is held as {1,0,0}, ie 2^1 * 3^0 * 5^0, and 15 as {0,1,1}, ie 2^0 * 3^1 * 5^1.

Notice that 2, being a power of 2, has zero in all slots other than 2.
We can say that (eg) 15 is not exactly divisible by 2 because an sq_sub({0,1,1},{1,0,0}) would contain some <0s.
We can also say that 15 is not a power of 2 because there are non-0 in other than n[find(2,known_factors)].
Division (to whole integer) is performed simply by subtracting the corresponding powers, as above not possible if any would be negative.

without js -- 8s
--with javascript_semantics -- 52s!! (see note)
constant steps = 20,
primes = 45

sequence known_factors = {} -- nb: no specific order

function combine_factors(sequence n)
-- (inverse of as_primes)
atom res = 1
for i=1 to length(n) do
if n[i]!=0 then
res *= power(known_factors[i],n[i])
end if
end for
return res
end function

function as_primes(integer n)
-- eg as_primes(55) -> {5,11} -> indexes to known_factors
if n=1 then return {} end if
sequence pf = prime_factors(n,duplicates:=true)
sequence res = repeat(0,length(known_factors))
for i=1 to length(pf) do
integer k = find(pf[i],known_factors)
if k=0 then
known_factors = append(known_factors,pf[i])
res = append(res,1)
else
res[k] += 1
end if
end for
--  atom chk = combine_factors(res)
--  if chk!=n then ?9/0 end if
return res
end function

function parse(string s)
sequence res = split(s)
for i=1 to length(res) do
sequence sri = scanf(res[i],"%d/%d")
if length(sri)!=1 then ?9/0 end if  -- oops!
integer {{n,d}} = sri
res[i] = {as_primes(n),as_primes(d)}
end for
return res
end function

function step(sequence pgm, sequence n)
for pc=1 to length(pgm) do
sequence d = pgm[pc][2], res = n
--      sequence d = pgm[pc][2], res = deep_copy(n) -- (see timing note)
bool ok = true
for f=1 to length(d) do
integer df = d[f]
if df!=0 then
if f>length(n) or df>n[f] then
ok = false
exit
end if
res[f] -= df
end if
end for
if ok then
n = pgm[pc][1]
integer zf = length(n)-length(res)
if zf>0 then res &= repeat(0,zf) end if
for i=1 to length(n) do
res[i] += n[i]
end for
return res
end if
end for
return 0
end function

constant src = "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"
sequence pgm = parse(src)
object n = as_primes(2)
sequence res = {}
for i=1 to steps do
n = step(pgm,n)
if n=0 then exit end if
res = append(res,combine_factors(n))
end for
printf(1,"first %d results: %v\n",{steps,res})

n = as_primes(2)
integer k2 = find(2,known_factors)
sequence n0 = repeat(0,length(known_factors))
res = {}
integer iteration = 1
atom t0 = time()
while length(res)<primes do
n = step(pgm,n)
if n=0 then exit end if
n0[k2] = n[k2]
if n=n0 then -- (ie all non-2 are 0)
-- and the prime itself is ready and waiting...
res = append(res,n[k2])
end if
iteration += 1
end while
printf(1,"first %d primes: %v\n",{primes,res})
printf(1,"%,d iterations in %s\n",{iteration,elapsed(time()-t0)})

Output:
first 20 results: {15,825,725,1925,2275,425,390,330,290,770,910,170,156,132,116,308,364,68,4,30}
first 45 primes: {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,
107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197}
10,494,775 iterations in 8.4s


For comparison with that 8.4s, I've collected Python: 386s, REXX: 60s for 25 (on tio, then it timed out), Ruby: 187s, Go: 1736s(!!), Julia 616s, FreeBasic 11.7s
(Clearly the algorithms being used are way more significant than any inherent programming language differences here.)
Update 17/01/2022: Also note that 8.4s is for desktop/Phix, without js. Adding with js and the required deep_copy() makes it about 6.5 times slower, though it actually fares somewhat better(/less awful) in a browser (30s vs 50s). Clearly the current hll implementation of deep_copy(), as first shipped in July 2021, is nowhere near as efficient as the older low-level "clone that must be avoided", at least not yet.

## Prolog

load(Program, Fractions) :-
re_split("[ ]+", Program, Split), odd_items(Split, TextualFractions),
maplist(convert_frac, TextualFractions, Fractions).

odd_items(L, L) :- L = [_], !.  % remove the even elements from a list.
odd_items([X,_|L], [X|R]) :- odd_items(L, R).

convert_frac(Text, Frac) :-
re_matchsub("([0-9]+)/([0-9]+)"/t, Text, Match, []),
Frac is Match.1 rdiv Match.2.

step(_, [], stop) :- !.
step(N, [F|Fs], R) :-
A is N*F,
(integer(A) -> R = A; step(N, Fs, R)).

exec(Prg, Start, Lz) :-
lazy_list(transition, Prg/Start, Lz).

transition(Prg/N0, Prg/N1, N1) :-
step(N0, Prg, N1).

steps(K, Start, Prg, Seq) :-
exec(Prg, Start, Values),
length(Seq, K), Seq = [Start|Rest], prefix(Rest, Values), !.

% The actual PRIMEGEN program follows...

primegen(Prg) :-
load("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", Prg).

primes(N, Primes) :-
primegen(Prg), exec(Prg, 2, Steps),
length(Primes, N), capture_primes(Primes, Steps).

capture_primes([], _) :- !.
capture_primes([P|Ps], [Q|Qs]) :- pow2(Q), !, P is lsb(Q), capture_primes(Ps, Qs).
capture_primes(Ps, [_|Qs]) :- capture_primes(Ps, Qs).

pow2(X) :- X /\ (X-1) =:= 0.

main :-
primegen(Prg), steps(15, 2, Prg, Steps),
format("The first 15 steps from PRIMEGEN are: ~w~n", [Steps]),
primes(20, Primes),
format("By running PRIMEGEN we found these primes: ~w~n", [Primes]),
halt.

?- main.

Output:
The first 15 steps from PRIMEGEN are: [2,15,825,725,1925,2275,425,390,330,290,770,910,170,156,132]
By running PRIMEGEN we found these primes: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71]


## Python

### Python: Generate series from a fractran program

from fractions import Fraction

def fractran(n, fstring='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'):
flist = [Fraction(f) for f in fstring.replace(' ', '').split(',')]

n = Fraction(n)
while True:
yield n.numerator
for f in flist:
if (n * f).denominator == 1:
break
else:
break
n *= f

if __name__ == '__main__':
n, m = 2, 15
print('First %i members of fractran(%i):\n  ' % (m, n) +
', '.join(str(f) for f,i in zip(fractran(n), range(m))))

Output:
First 15 members of fractran(2):
2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132

#### Python: Generate primes

Use fractran above as a module imported into the following program.

from fractran import fractran

def fractran_primes():
for i, fr in enumerate(fractran(2), 1):
binstr = bin(fr)[2:]
if binstr.count('1') == 1:
prime = binstr.count('0')
if prime > 1:   # Skip 2**0 and 2**1
yield prime, i

if __name__ == '__main__':
for (prime, i), j in zip(fractran_primes(), range(15)):
print("Generated prime %2i from the %6i'th member of the fractran series" % (prime, i))

Output:
Generated prime  2 from the     20'th member of the fractran series
Generated prime  3 from the     70'th member of the fractran series
Generated prime  5 from the    281'th member of the fractran series
Generated prime  7 from the    708'th member of the fractran series
Generated prime 11 from the   2364'th member of the fractran series
Generated prime 13 from the   3877'th member of the fractran series
Generated prime 17 from the   8069'th member of the fractran series
Generated prime 19 from the  11320'th member of the fractran series
Generated prime 23 from the  19202'th member of the fractran series
Generated prime 29 from the  36867'th member of the fractran series
Generated prime 31 from the  45552'th member of the fractran series
Generated prime 37 from the  75225'th member of the fractran series
Generated prime 41 from the 101113'th member of the fractran series
Generated prime 43 from the 117832'th member of the fractran series
Generated prime 47 from the 152026'th member of the fractran series

## Quackery

run performs the first step of executing a Fractran program, returning a rational number (represented as two stack items: numerator and denominator) and a boolean. The boolean is true if the halting condition is satisfied, and false otherwise.

In this task the parsed Fractran program is stored on the ancillary stack program.

To execute a Fractran program until the halting condition is satisfied, use [ program share run until ]. The Fractran prime generator will never satisfy the halting condition, so in this task the drop after run discards the boolean.

  [ $"bigrat.qky" loadfile ] now! [ 1 & not ] is even ( n --> b ) [ nip 1 = ] is vint ( n/d --> b ) [ [ dup even while 1 >> again ] 1 = ] is powerof2 ( n --> b ) [ 0 swap [ dup even while dip 1+ 1 >> again ] drop ] is lowbit ( n --> n ) [ [] swap nest$
witheach
[ char / over find
space unrot poke
build nested join ] ]  is parse$($ --> [     )

[ stack ]                    is program  (     s -->       )

[ true temp put
witheach
[ do 2over v*
2dup vint iff
[ false temp replace
conclude ]
else 2drop ]
2swap 2drop
temp take ]              is run      ( n/d [ --> n/d b )

[ stack ]                    is primes   (       --> s     )

$"17/91 78/85 19/51 23/38 29/33 77/29 95/23"$ " 77/19 1/17 11/13 13/11 15/14 15/2 55/1" join
parse$program put 2 n->v 15 times [ program share run drop over echo sp ] cr 2drop 2 n->v [] primes put [ program share run drop over dup powerof2 iff [ lowbit primes take swap join primes put ] else drop primes share size 20 = until ] 2drop primes take echo program release Output: 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 [ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 ]  ## Racket Translation of: D Simple version, without sequences. #lang racket (define (displaysp x) (display x) (display " ")) (define (read-string-list str) (map string->number (string-split (string-replace str " " "") ","))) (define (eval-fractran n list) (for/or ([e (in-list list)]) (let ([en (* e n)]) (and (integer? en) en)))) (define (show-fractran fr n s) (printf "First ~a members of fractran(~a):\n" s n) (displaysp n) (for/fold ([n n]) ([i (in-range (- s 1))]) (let ([new-n (eval-fractran n fr)]) (displaysp new-n) new-n)) (void)) (define fractran (read-string-list (string-append "17 / 91, 78 / 85, 19 / 51, 23 / 38, 29 / 33," "77 / 29, 95 / 23, 77 / 19, 1 / 17, 11 / 13," "13 / 11, 15 / 14, 15 / 2, 55 / 1"))) (show-fractran fractran 2 15)  Output: First 15 members of fractran(2): 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 ## Raku (formerly Perl 6) Works with: rakudo version 2015-11-03 A Fractran program potentially returns an infinite list, and infinite lists are a common data structure in Raku. The limit is therefore enforced only by slicing the infinite list. sub fractran(@program) { 2, { first Int, map (* *$_).narrow, @program } ... 0
}
say fractran(<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>)[^100];

Output:
(2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4 30 225 12375 10875 28875 25375 67375 79625 14875 13650 2550 2340 1980 1740 4620 4060 10780 12740 2380 2184 408 152 92 380 230 950 575 2375 9625 11375 2125 1950 1650 1450 3850 4550 850 780 660 580 1540 1820 340 312 264 232 616 728 136 8 60 450 3375 185625 163125 433125 380625 1010625 888125 2358125 2786875 520625 477750 89250 81900 15300 14040 11880 10440 27720 24360 64680 56840 150920 178360 33320 30576 5712 2128 1288)

Extra credit: We can weed out all the powers of two into another infinite constant list based on the first list. In this case the sequence is limited only by our patience, and a ^C from the terminal. The .msb method finds the most significant bit of an integer, which conveniently is the base-2 log of the power-of-two in question.

sub fractran(@program) {
2, { first Int, map (* * $_).narrow, @program } ... 0 } for fractran <17/91 78/85 19/51 23/38 29/33 77/29 95/23 77/19 1/17 11/13 13/11 15/14 15/2 55/1> { say$++, "\t", .msb, "\t", $_ if 1 +< .msb ==$_;
}

Output:
0       1       2
1	2	4
2	3	8
3	5	32
4	7	128
5	11	2048
6	13	8192
7	17	131072
8	19	524288
9	23	8388608
^C

## Red

Red ["Fractran"]

inp: ask "please enter list of fractions, or input file name: "
if exists? inpf: to-file inp [inp: read inpf]

digit: charset "0123456789"
frac: [copy p [some digit] #"/" copy q [some digit]
keep (as-pair to-integer p to-integer q)]
code: parse inp [collect [frac some [[some " "] frac]]]

x: to-integer ask "please enter the number of terms, hit return for no limit: "
l: length? code
loop x [
forall code [
c: code/1
if n % c/y = 0 [
print n: n / c/y * c/x
break
]
if l = index? code [halt]
]
]

Output:
please enter list of fractions, or input file name: 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
please enter starting number n: 2
please enter the number of terms, hit return for no limit: 15
15
825
725
1925
2275
425
390
330
290
770
910
170
156
132
116

## REXX

Programming note:   extra blanks can be inserted in the fractions before and/or after the solidus   [/].

### showing all terms

/*REXX program runs  FRACTRAN  for a given set of  fractions  and  from a specified  N. */
numeric digits 2000                              /*be able to handle larger numbers.    */
parse arg N terms fracs                          /*obtain optional arguments from the CL*/
if     N=='' |     N==","  then     N=   2       /*Not specified?  Then use the default.*/
if terms=='' | terms==","  then terms= 100       /* "      "         "   "   "      "   */
if fracs=''                then fracs= "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'
/* [↑]  The default for the fractions. */
f= space(fracs, 0)                               /*remove all blanks from the FRACS list*/
do #=1  while f\=='';    parse var  f   n.#   "/"   d.#   ','   f
end   /*#*/                  /* [↑]  parse all the fractions in list*/
#= # - 1                                         /*the number of fractions just found.  */
say #   'fractions:'   fracs                     /*display number and actual fractions. */
say 'N  is starting at '   N                     /*display the starting number  N.      */
say terms   ' terms are being shown:'            /*display a kind of header/title.      */

do    j=1  for  terms                        /*perform the DO loop for each   term. */
do k=1  for  #                            /*   "     "   "   "   "    "  fraction*/
if N // d.k \== 0  then iterate           /*Not an integer?  Then ignore it.     */
cN= commas(N);     L= length(cN)          /*maybe insert commas into N;  get len.*/
say right('term' commas(j), 44) "──► " right(cN, max(15, L))  /*show Nth term & N*/
N= N  %  d.k  *  n.k                      /*calculate next term (use %≡integer ÷)*/
leave                                     /*go start calculating the next term.  */
end   /*k*/                               /* [↑]  if an integer, we found a new N*/
end      /*j*/
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?

output   when using the default input:
14 fractions: 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
N  is starting at  2
100  terms are being shown:
term 1 ──►                2
term 2 ──►               15
term 3 ──►              825
term 4 ──►              725
term 5 ──►            1,925
term 6 ──►            2,275
term 7 ──►              425
term 8 ──►              390
term 9 ──►              330
term 10 ──►              290
term 11 ──►              770
term 12 ──►              910
term 13 ──►              170
term 14 ──►              156
term 15 ──►              132
term 16 ──►              116
term 17 ──►              308
term 18 ──►              364
term 19 ──►               68
term 20 ──►                4
term 21 ──►               30
term 22 ──►              225
term 23 ──►           12,375
term 24 ──►           10,875
term 25 ──►           28,875
term 26 ──►           25,375
term 27 ──►           67,375
term 28 ──►           79,625
term 29 ──►           14,875
term 30 ──►           13,650
term 31 ──►            2,550
term 32 ──►            2,340
term 33 ──►            1,980
term 34 ──►            1,740
term 35 ──►            4,620
term 36 ──►            4,060
term 37 ──►           10,780
term 38 ──►           12,740
term 39 ──►            2,380
term 40 ──►            2,184
term 41 ──►              408
term 42 ──►              152
term 43 ──►               92
term 44 ──►              380
term 45 ──►              230
term 46 ──►              950
term 47 ──►              575
term 48 ──►            2,375
term 49 ──►            9,625
term 50 ──►           11,375
term 51 ──►            2,125
term 52 ──►            1,950
term 53 ──►            1,650
term 54 ──►            1,450
term 55 ──►            3,850
term 56 ──►            4,550
term 57 ──►              850
term 58 ──►              780
term 59 ──►              660
term 60 ──►              580
term 61 ──►            1,540
term 62 ──►            1,820
term 63 ──►              340
term 64 ──►              312
term 65 ──►              264
term 66 ──►              232
term 67 ──►              616
term 68 ──►              728
term 69 ──►              136
term 70 ──►                8
term 71 ──►               60
term 72 ──►              450
term 73 ──►            3,375
term 74 ──►          185,625
term 75 ──►          163,125
term 76 ──►          433,125
term 77 ──►          380,625
term 78 ──►        1,010,625
term 79 ──►          888,125
term 80 ──►        2,358,125
term 81 ──►        2,786,875
term 82 ──►          520,625
term 83 ──►          477,750
term 84 ──►           89,250
term 85 ──►           81,900
term 86 ──►           15,300
term 87 ──►           14,040
term 88 ──►           11,880
term 89 ──►           10,440
term 90 ──►           27,720
term 91 ──►           24,360
term 92 ──►           64,680
term 93 ──►           56,840
term 94 ──►          150,920
term 95 ──►          178,360
term 96 ──►           33,320
term 97 ──►           30,576
term 98 ──►            5,712
term 99 ──►            2,128
term 100 ──►            1,288


### showing prime numbers

Programming note:   if the number of terms specified (the 2nd argument) is negative, then only powers of two are displayed.

/*REXX program runs  FRACTRAN  for a given set of  fractions  and  from a specified  N. */
numeric digits 999;  d= digits();   w= length(d) /*be able to handle gihugeic numbers.  */
parse arg N terms fracs                          /*obtain optional arguments from the CL*/
if     N=='' |     N==","  then     N=   2       /*Not specified?  Then use the default.*/
if terms=='' | terms==","  then terms= 100       /* "      "         "   "   "      "   */
if fracs=''                then fracs= "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'
/* [↑]  The default for the fractions. */
f= space(fracs, 0)                               /*remove all blanks from the FRACS list*/
do #=1  while f\=='';    parse var  f    n.#   "/"   d.#   ','   f
end   /*#*/                  /* [↑]  parse all the fractions in list*/
#= # - 1                                         /*adjust the number of fractions found.*/
tell= terms>0                                    /*flag:  show number  or  a power of 2.*/
!.= 0;                              _= 1         /*the default value  for powers of  2. */
if \tell  then do p=1  until length(_)>d;                 _= _ + _;              !._= 1
if p==1  then @._= left('', w + 9)        "2**"left(p, w)    ' '
else @._= '(prime' right(p, w)")  2**"left(p, w)    ' '
end   /*p*/                       /* [↑]  build   powers of 2   tables.  */
L= length(N)                                     /*length in decimal digits of integer N*/
say #  'fractions:'  fracs                       /*display number and actual fractions. */
say 'N  is starting at '  N                      /*display the starting number   N.     */
if tell  then say terms  ' terms are being shown:'                     /*display header.*/
else say 'only powers of two are being shown:'                /*   "       "   */
@a= '(max digits used:'                          /*a literal used in the   SAY   below. */

do j=1  for  abs(terms)                        /*perform DO loop once for each  term. */
do k=1  for  #                              /*   "     "   "    "   "    " fraction*/
if N // d.k \== 0  then iterate             /*Not an integer?  Then ignore it.     */
cN= commas(N);                 cj=commas(j) /*maybe insert commas into  N.         */
if tell  then say right('term' cj, 44)   "──► "   cN      /*display Nth term and N.*/
else if !.N  then say right('term' cj,15)  "──►"  @.N @a  right(L,w)")  " cN
N= N  %  d.k  *  n.k                        /*calculate next term (use %≡integer ÷)*/
L= max(L, length(N) )                       /*the maximum number of decimal digits.*/
leave                                       /*go start calculating the next term.  */
end   /*k*/                                 /* [↑]  if an integer, we found a new N*/
end      /*j*/
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?

output   when using the input (negative fifty million) of:     ,   -50000000
14 fractions: 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
N  is starting at  2
only powers of two are being shown:
term 1 ──►              2**1     (max digits used:   1)   2
term 20 ──► (prime   2)  2**2     (max digits used:   4)   4
term 70 ──► (prime   3)  2**3     (max digits used:   5)   8
term 281 ──► (prime   5)  2**5     (max digits used:   8)   32
term 708 ──► (prime   7)  2**7     (max digits used:  12)   128
term 2,364 ──► (prime  11)  2**11    (max digits used:  18)   2,048
term 3,877 ──► (prime  13)  2**13    (max digits used:  21)   8,192
term 8,069 ──► (prime  17)  2**17    (max digits used:  27)   131,072
term 11,320 ──► (prime  19)  2**19    (max digits used:  30)   524,288
term 19,202 ──► (prime  23)  2**23    (max digits used:  36)   8,388,608
term 36,867 ──► (prime  29)  2**29    (max digits used:  46)   536,870,912
term 45,552 ──► (prime  31)  2**31    (max digits used:  49)   2,147,483,648
term 75,225 ──► (prime  37)  2**37    (max digits used:  58)   137,438,953,472
term 101,113 ──► (prime  41)  2**41    (max digits used:  64)   2,199,023,255,552
term 117,832 ──► (prime  43)  2**43    (max digits used:  67)   8,796,093,022,208
term 152,026 ──► (prime  47)  2**47    (max digits used:  73)   140,737,488,355,328
term 215,385 ──► (prime  53)  2**53    (max digits used:  83)   9,007,199,254,740,992
term 293,376 ──► (prime  59)  2**59    (max digits used:  92)   576,460,752,303,423,488
term 327,021 ──► (prime  61)  2**61    (max digits used:  95)   2,305,843,009,213,693,952
term 428,554 ──► (prime  67)  2**67    (max digits used: 104)   147,573,952,589,676,412,928
term 507,520 ──► (prime  71)  2**71    (max digits used: 110)   2,361,183,241,434,822,606,848
term 555,695 ──► (prime  73)  2**73    (max digits used: 113)   9,444,732,965,739,290,427,392
term 700,064 ──► (prime  79)  2**79    (max digits used: 123)   604,462,909,807,314,587,353,088
term 808,332 ──► (prime  83)  2**83    (max digits used: 129)   9,671,406,556,917,033,397,649,408
term 989,527 ──► (prime  89)  2**89    (max digits used: 138)   618,970,019,642,690,137,449,562,112
term 1,273,491 ──► (prime  97)  2**97    (max digits used: 151)   158,456,325,028,528,675,187,087,900,672
term 1,434,367 ──► (prime 101)  2**101   (max digits used: 157)   2,535,301,200,456,458,802,993,406,410,752
term 1,530,214 ──► (prime 103)  2**103   (max digits used: 160)   10,141,204,801,825,835,211,973,625,643,008
term 1,710,924 ──► (prime 107)  2**107   (max digits used: 166)   162,259,276,829,213,363,391,578,010,288,128
term 1,818,255 ──► (prime 109)  2**109   (max digits used: 169)   649,037,107,316,853,453,566,312,041,152,512
term 2,019,963 ──► (prime 113)  2**113   (max digits used: 175)   10,384,593,717,069,655,257,060,992,658,440,192
term 2,833,090 ──► (prime 127)  2**127   (max digits used: 197)   170,141,183,460,469,231,731,687,303,715,884,105,728
term 3,104,686 ──► (prime 131)  2**131   (max digits used: 203)   2,722,258,935,367,507,707,706,996,859,454,145,691,648
term 3,546,321 ──► (prime 137)  2**137   (max digits used: 212)   174,224,571,863,520,493,293,247,799,005,065,324,265,472
term 3,720,786 ──► (prime 139)  2**139   (max digits used: 215)   696,898,287,454,081,973,172,991,196,020,261,297,061,888
term 4,549,719 ──► (prime 149)  2**149   (max digits used: 231)   713,623,846,352,979,940,529,142,984,724,747,568,191,373,312
term 4,755,582 ──► (prime 151)  2**151   (max digits used: 234)   2,854,495,385,411,919,762,116,571,938,898,990,272,765,493,248
term 5,329,875 ──► (prime 157)  2**157   (max digits used: 243)   182,687,704,666,362,864,775,460,604,089,535,377,456,991,567,872
term 5,958,404 ──► (prime 163)  2**163   (max digits used: 252)   11,692,013,098,647,223,345,629,478,661,730,264,157,247,460,343,808
term 6,400,898 ──► (prime 167)  2**167   (max digits used: 259)   187,072,209,578,355,573,530,071,658,587,684,226,515,959,365,500,928
term 7,120,509 ──► (prime 173)  2**173   (max digits used: 268)   11,972,621,413,014,756,705,924,586,149,611,790,497,021,399,392,059,392
term 7,868,448 ──► (prime 179)  2**179   (max digits used: 277)   766,247,770,432,944,429,179,173,513,575,154,591,809,369,561,091,801,088
term 8,164,153 ──► (prime 181)  2**181   (max digits used: 280)   3,064,991,081,731,777,716,716,694,054,300,618,367,237,478,244,367,204,352
term 9,541,986 ──► (prime 191)  2**191   (max digits used: 296)   3,138,550,867,693,340,381,917,894,711,603,833,208,051,177,722,232,017,256,448
term 9,878,163 ──► (prime 193)  2**193   (max digits used: 299)   12,554,203,470,773,361,527,671,578,846,415,332,832,204,710,888,928,069,025,792
term 10,494,775 ──► (prime 197)  2**197   (max digits used: 305)   200,867,255,532,373,784,442,745,261,542,645,325,315,275,374,222,849,104,412,672
term 10,852,158 ──► (prime 199)  2**199   (max digits used: 308)   803,469,022,129,495,137,770,981,046,170,581,301,261,101,496,891,396,417,650,688
term 12,871,594 ──► (prime 211)  2**211   (max digits used: 327)   3,291,009,114,642,412,084,309,938,365,114,701,009,965,471,731,267,159,726,697,218,048
term 15,137,114 ──► (prime 223)  2**223   (max digits used: 345)   13,479,973,333,575,319,897,333,507,543,509,815,336,818,572,211,270,286,240,551,805,124,608
term 15,956,646 ──► (prime 227)  2**227   (max digits used: 351)   215,679,573,337,205,118,357,336,120,696,157,045,389,097,155,380,324,579,848,828,881,993,728
term 16,429,799 ──► (prime 229)  2**229   (max digits used: 354)   862,718,293,348,820,473,429,344,482,784,628,181,556,388,621,521,298,319,395,315,527,974,912
term 17,293,373 ──► (prime 233)  2**233   (max digits used: 361)   13,803,492,693,581,127,574,869,511,724,554,050,904,902,217,944,340,773,110,325,048,447,598,592
term 18,633,402 ──► (prime 239)  2**239   (max digits used: 370)   883,423,532,389,192,164,791,648,750,371,459,257,913,741,948,437,809,479,060,803,100,646,309,888
term 19,157,411 ──► (prime 241)  2**241   (max digits used: 373)   3,533,694,129,556,768,659,166,595,001,485,837,031,654,967,793,751,237,916,243,212,402,585,239,552
term 21,564,310 ──► (prime 251)  2**251   (max digits used: 388)   3,618,502,788,666,131,106,986,593,281,521,497,120,414,687,020,801,267,626,233,049,500,247,285,301,248
term 23,157,731 ──► (prime 257)  2**257   (max digits used: 398)   231,584,178,474,632,390,847,141,970,017,375,815,706,539,969,331,281,128,078,915,168,015,826,259,279,872
term 24,805,778 ──► (prime 263)  2**263   (max digits used: 407)   14,821,387,422,376,473,014,217,086,081,112,052,205,218,558,037,201,992,197,050,570,753,012,880,593,911,808
term 26,506,125 ──► (prime 269)  2**269   (max digits used: 416)   948,568,795,032,094,272,909,893,509,191,171,341,133,987,714,380,927,500,611,236,528,192,824,358,010,355,712
term 27,168,588 ──► (prime 271)  2**271   (max digits used: 419)   3,794,275,180,128,377,091,639,574,036,764,685,364,535,950,857,523,710,002,444,946,112,771,297,432,041,422,848
term 28,973,145 ──► (prime 277)  2**277   (max digits used: 428)   242,833,611,528,216,133,864,932,738,352,939,863,330,300,854,881,517,440,156,476,551,217,363,035,650,651,062,272
term 30,230,537 ──► (prime 281)  2**281   (max digits used: 435)   3,885,337,784,451,458,141,838,923,813,647,037,813,284,813,678,104,279,042,503,624,819,477,808,570,410,416,996,352
term 30,952,920 ──► (prime 283)  2**283   (max digits used: 438)   15,541,351,137,805,832,567,355,695,254,588,151,253,139,254,712,417,116,170,014,499,277,911,234,281,641,667,985,408
term 34,284,307 ──► (prime 293)  2**293   (max digits used: 453)   15,914,343,565,113,172,548,972,231,940,698,266,883,214,596,825,515,126,958,094,847,260,581,103,904,401,068,017,057,792
term 39,303,996 ──► (prime 307)  2**307   (max digits used: 475)   260,740,604,970,814,219,042,361,048,116,400,404,614,587,954,389,239,840,081,425,977,517,360,806,369,707,098,391,474,864,128
term 40,844,960 ──► (prime 311)  2**311   (max digits used: 481)   4,171,849,679,533,027,504,677,776,769,862,406,473,833,407,270,227,837,441,302,815,640,277,772,901,915,313,574,263,597,826,048
term 41,728,501 ──► (prime 313)  2**313   (max digits used: 484)   16,687,398,718,132,110,018,711,107,079,449,625,895,333,629,080,911,349,765,211,262,561,111,091,607,661,254,297,054,391,304,192
term 43,329,639 ──► (prime 317)  2**317   (max digits used: 490)   266,998,379,490,113,760,299,377,713,271,194,014,325,338,065,294,581,596,243,380,200,977,777,465,722,580,068,752,870,260,867,072
term 49,260,306 ──► (prime 331)  2**331   (max digits used: 512)   4,374,501,449,566,023,848,745,004,454,235,242,730,706,338,861,786,424,872,851,541,212,819,905,998,398,751,846,447,026,354,046,107,648
...
(some output elided.)
...
term 193,455,490 ──► (prime 523)  2**523   (max digits used: 808)   27,459,190,640,522,438,859,927,603,196,325,572,869,077,741,200,573,221,637,577,853,836,742,172,733,590,624,208,490,238,562,645,818,219,909,185,245,565,923,432,148,487,951,998,866,575,250,296,113,164,460,228,608


Output note:   There are intermediary numbers (that aren't powers of two) that are hundreds of digits long.

## RPL

≪ → text
≪ "{'" 1 text SIZE FOR j
text j DUP SUB
IF DUP " " == THEN DROP "' '" END +
NEXT
"'}" + STR→
≫ ≫ 'PRECOMPIL' STO     @ ( "fractions" → { 'fractions' } )

≪ SWAP 20 → prog steps
≪ {} SWAP
1 steps FOR n
1 CF
1 prog SIZE FOR j
prog j GET OVER * EVAL RND
IF DUP FP THEN DROP ELSE
SWAP DROP SWAP OVER + SWAP
prog SIZE 'j' STO 1 SF END
NEXT
IF 1 FC?C THEN steps 'n' STO END
NEXT DROP
≫ ≫ 'RUN20' STO         @ ( { 'fractions' } n → { results } )

"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" PRECOMPIL
2 RUN20

Output:
1: { 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4 30 }


## Ruby

ar = %w[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]
FractalProgram = ar.map(&:to_r)                                #=> array of rationals

Runner = Enumerator.new do |y|
num = 2
loop{ y << num *= FractalProgram.detect{|f| (num*f).denominator == 1} }
end

prime_generator = Enumerator.new do |y|
Runner.each do |num|
l = Math.log2(num)
y << l.to_i if l.floor == l
end
end

# demo
p Runner.take(20).map(&:numerator)
p prime_generator.take(20)

Output:
[15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4, 30]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]


## Scala

class TestFractran extends FunSuite {
val program = Fractran("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")
val expect = List(2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132)

test("find first fifteen fractran figures") {
assert((program .execute(2) take 15 toList) === expect)
}
}

object Fractran {
val pattern = """\s*(\d+)\s*/\s*(\d+)\s*""".r
def parse(m: Match) = ((m group 1).toInt, (m group 2).toInt)
def apply(program: String) =  new Fractran(
pattern.findAllMatchIn(program).map(parse).toList)
}

class Fractran(val numDem: List[(Int,Int)]) {
def execute(value: Int) = unfold(value) { v =>
numDem indexWhere(v % _._2 == 0) match {
case i if i > -1 => Some(v, numDem(i)._1 * v / numDem(i)._2)
case _ => None
}
}
}


## Scheme

Scheme naturally handles fractions, translating to integers as required. The first part of the code translates from a string representation, as required, but equally the user could type the list of fractions in directly as a list.

Library: Scheme/SRFIs

Similar to Python implementation of generating primes, the power of 2 is detected by first converting the number to binary representation, and check if it has only 1 "1" bit.

(import (scheme base)
(scheme inexact)
(scheme write)
(srfi 13))   ;; for string-length and string-ref

(define *string-fractions* ; string input of fractions
"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")

(define *fractions* ; create vector of fractions from string input
(list->vector ; convert result to a vector, for constant access times
(read (open-input-string ; read from the string of fractions, as a list
(string-append "(" *string-fractions* ")")))))

;; run a fractran interpreter, returning the next number for n
;; or #f if no next number available
;; assume fractions: ordered vector of positive fractions
;;                n: a positive integer
(define (fractran fractions n)
(let ((max-n (vector-length fractions)))
(let loop ((i 0))
(cond ((= i max-n)
#f)
((integer? (* n (vector-ref fractions i)))
(* n (vector-ref fractions i)))
(else
(loop (+ 1 i)))))))

(define (display-result max-n)
(do ((i 0 (+ 1 i))
(n 2 (fractran *fractions* n)))
((= i max-n) (newline))
(display n) (display " ")))

(display-result 20) ; show first 20 numbers

;; Extra Credit: derive first 20 prime numbers
(define (generate-primes target-number initial-n)
(define (is-power-of-two? n) ; a binary with only 1 "1" bit is a power of 2
(cond ((<= n 2) ; exclude 2 and 1
#f)
(else
(let loop ((i 0) (acc 0) (binary-str (number->string n 2)))
(cond ((= i (string-length binary-str))
#t)
((and (eq? (string-ref binary-str i) #\1) (= 1 acc))
#f)
((eq? (string-ref binary-str i) #\1)
(loop (+ 1 i) (+ 1 acc) binary-str))
(else
(loop (+ 1 i) acc binary-str)))))))
(define (extract-prime n) ; just gets the number of zeroes in binary
(let ((binary-str (number->string n 2)))
(- (string-length binary-str) 1)))
;
(let loop ((count 0)
(n initial-n))
(when (< count target-number)
(cond ((eq? n #f)
(display "-- FAILED TO COMPUTE N --\n"))
((is-power-of-two? n)
(display (extract-prime n)) (display " ")
(loop (+ 1 count)
(fractran *fractions* n)))
(else
(loop count
(fractran *fractions* n))))))
(newline))

(display "Primes:\n")
(generate-primes 20 2) ; create first 20 primes

Output:
Task: 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4
Primes:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71


## Seed7

$include "seed7_05.s7i"; include "rational.s7i"; const func array integer: fractran (in integer: limit, in var integer: number, in array rational: program) is func result var array integer: output is 0 times 0; local var integer: index is 1; var rational: newNumber is 0/1; begin output := [] (number); while index <= length(program) and length(output) <= limit do newNumber := rat(number) * program[index]; if newNumber = rat(trunc(newNumber)) then number := trunc(newNumber); output &:= number; index := 1; else incr(index); end if; end while; end func; const proc: main is func local const array rational: program is [] (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); var array integer: output is 0 times 0; var integer: number is 0; begin output := fractran(15, 2, program); for number range output do write(number <& " "); end for; writeln; end func; Output: 2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116  Program to compute prime numbers with fractran (The program has no limit, use CTRL-C to terminate it): $ include "seed7_05.s7i";
include "bigrat.s7i";

const proc: fractran (in var bigInteger: number, in array bigRational: program) is func
local
var integer: index is 1;
var bigRational: newNumber is 0_/1_;
begin
while index <= length(program) do
newNumber := rat(number) * program[index];
if newNumber = rat(trunc(newNumber)) then
number := trunc(newNumber);
if 2_ ** ord(log2(number)) = number then
writeln(log2(number));
end if;
index := 1;
else
incr(index);
end if;
end while;
end func;

const proc: main is func
local
const array bigRational: program is []
(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_);
begin
fractran(2_, program);
end func;
Output:
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101


## Sidef

Translation of: Ruby
var str ="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"
const FractalProgram = str.split(',').map{.num}      #=> array of rationals

func runner(n, callback) {
var num = 2
n.times {
callback(num *= FractalProgram.find { |f| f * num -> is_int })
}
}

func prime_generator(n, callback) {
var x = 0;
runner(Inf, { |num|
var l = num.log2
if (l.floor == l) {
callback(l.int)
++x == n && return nil
}
})
}

STDOUT.autoflush(true)

runner(20, {|n| print (n, ' ') })
print "\n"

prime_generator(20, {|n| print (n, ' ') })
print "\n"

Output:
15 825 725 1925 2275 425 390 330 290 770 910 170 156 132 116 308 364 68 4 30
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71


## Tcl

Works with: Tcl version 8.6
package require Tcl 8.6

oo::class create Fractran {
variable fracs nco
constructor {fractions} {
set fracs {}
foreach frac $fractions { if {[regexp {^(\d+)/(\d+),?$} $frac -> num denom]} { lappend fracs$num $denom } else { return -code error "$frac is not a supported fraction"
}
}
if {![llength $fracs]} { return -code error "need at least one fraction" } } method execute {n {steps 15}} { set co [coroutine [incr nco] my Generate$n]
for {set i 0} {$i <$steps} {incr i} {
lappend result [$co] } catch {rename$co ""}
return $result } method Step {n} { foreach {num den}$fracs {
if {$n %$den} continue
return [expr {$n *$num / $den}] } return -code break } method Generate {n} { yield [info coroutine] while 1 { yield$n
set n [my Step $n] } return -code break } } set ft [Fractran new { 17/91 78/85 19/51 23/38 29/33 77/29 95/23 77/19 1/17 11/13 13/11 15/14 15/2 55/1 }] puts [$ft execute 2]

Output:
2 15 825 725 1925 2275 425 390 330 290 770 910 170 156 132


You can just collect powers of 2 by monkey-patching in something like this:

oo::objdefine $ft method pow2 {n} { set co [coroutine [incr nco] my Generate 2] set pows {} while {[llength$pows] < $n} { set item [$co]
if {($item & ($item-1)) == 0} {
lappend pows $item } } return$pows
}
puts [\$ft pow2 10]


Which will then produce this additional output:

2 4 8 32 128 2048 8192 131072 524288 8388608


## TI-83 BASIC

Works with: TI-83 BASIC version TI-84Plus 2.55MP
100->T
2->N
{17,78,19,23,29,77,95,77, 1,11,13,15,15,55}->LA
{91,85,51,38,33,29,23,19,17,13,11,14, 2, 1}->LB
Dim(LA)->U
T->Dim(LC)
For(I,1,T)
1->J: 1->F
While J<=U and F=1
If remainder(N,LB(J))=0
Then
Disp N
N->LC(I)
iPart(N/LB(J))*LA(J)->N
0->F
End
J+1->J
End
End

Note:

 -> stands for Store symbol
L  stands for List  symbol in LA,LB,LC

Output:
           2
15
825
725
1925
2275
425
390
330
290
...
2128
1288


## VBA

This implementations follows the Wikipedia description of FRACTRAN. There are test, decrement and increment instructions on an array of variables which are the exponents of the prime factors of the argument, which are the only instructions used to run the program with the function run, or go through it step by step with the function steps. An auxiliary factor function is used to compile the FRACTRAN program.

Option Base 1
Public prime As Variant
Public nf As New Collection
Public df As New Collection
Const halt = 20
Private Sub init()
prime = [{2,3,5,7,11,13,17,19,23,29,31}]
End Sub
Private Function factor(f As Long) As Variant
Dim result(10) As Integer
Dim i As Integer: i = 1
Do While f > 1
Do While f Mod prime(i) = 0
f = f \ prime(i)
result(i) = result(i) + 1
Loop
i = i + 1
Loop
factor = result
End Function
Private Function decrement(ByVal a As Variant, b As Variant) As Variant
For i = LBound(a) To UBound(a)
a(i) = a(i) - b(i)
Next i
decrement = a
End Function
Private Function increment(ByVal a As Variant, b As Variant) As Variant
For i = LBound(a) To UBound(a)
a(i) = a(i) + b(i)
Next i
increment = a
End Function
Private Function test(a As Variant, b As Variant)
flag = True
For i = LBound(a) To UBound(a)
If a(i) < b(i) Then
flag = False
Exit For
End If
Next i
test = flag
End Function
Private Function unfactor(x As Variant) As Long
result = 1
For i = LBound(x) To UBound(x)
result = result * prime(i) ^ x(i)
Next i
unfactor = result
End Function
Private Sub compile(program As String)
program = Replace(program, " ", "")
programlist = Split(program, ",")
For Each instruction In programlist
parts = Split(instruction, "/")
Next instruction
End Sub
Private Function run(x As Long) As Variant
n = factor(x)
counter = 0
Do While True
For i = 1 To df.Count
If test(n, df(i)) Then
n = increment(decrement(n, df(i)), nf(i))
Exit For
End If
Next i
Debug.Print unfactor(n);
counter = counter + 1
If num = 31 Or counter >= halt Then Exit Do
Loop
Debug.Print
run = n
End Function
Private Function steps(x As Variant) As Variant
'expects x=factor(n)
For i = 1 To df.Count
If test(x, df(i)) Then
x = increment(decrement(x, df(i)), nf(i))
Exit For
End If
Next i
steps = x
End Function
Private Function is_power_of_2(x As Variant) As Boolean
flag = True
For i = LBound(x) + 1 To UBound(x)
If x(i) > 0 Then
flag = False
Exit For
End If
Next i
is_power_of_2 = flag
End Function
Private Function filter_primes(x As Long, max As Integer) As Long
n = factor(x)
i = 0: iterations = 0
Do While i < max
If is_power_of_2(steps(n)) Then
Debug.Print n(1);
i = i + 1
End If
iterations = iterations + 1
Loop
Debug.Print
filter_primes = iterations
End Function
Public Sub main()
init
compile ("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")
Debug.Print "First 20 results:"
output = run(2)
Debug.Print "First 30 primes:"
Debug.Print "after"; filter_primes(2, 30); "iterations."
End Sub

Output:
First 20 results:
15  825  725  1925  2275  425  390  330  290  770  910  170  156  132  116  308  364  68  4  30
First 30 primes:
2  3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97  101  103  107  109  113
after 2019962 iterations.

## Wren

Translation of: Kotlin
Library: Wren-big

Extra credit is glacially slow. We just find the first 10 primes which takes about 85 seconds.

import "./big" for BigInt, BigRat

var isPowerOfTwo = Fn.new { |bi| bi & (bi - BigInt.one) == BigInt.zero }

var fractran = Fn.new { |program, n, limit, primesOnly|
var fractions = program.split(" ").where { |s| s != "" }
.map { |s| BigRat.fromRationalString(s) }
.toList
var results = []
var nn = BigInt.new(n)
while (results.count < limit) {
var fracs = fractions.where { |f| (f * nn).isInteger }.toList
if (fracs.count == 0) break
var frac = fracs[0]
nn = nn * frac.num / frac.den
if (!primesOnly) {
} else if (primesOnly && isPowerOfTwo.call(nn)) {
var prime = (nn.toNum.log / 2.log).floor
}
}
return results
}

var program = "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"
System.print("First twenty numbers:")
System.print(fractran.call(program, 2, 20, false))
System.print("\nFirst ten primes:")
System.print(fractran.call(program, 2, 10, true))

Output:
First twenty numbers:
[2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4]

First ten primes:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]


## zkl

var fracs="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";
fcn fractranW(n,fracsAsOneBigString){ //-->Walker (iterator)
fracs:=(fracsAsOneBigString-" ").split(",").apply(
fcn(frac){ frac.split("/").apply("toInt") }); //( (n,d), (n,d), ...)
Walker(fcn(rn,fracs){
n:=rn.value;
foreach a,b in (fracs){
if(n*a%b == 0){
rn.set(n*a/b);
return(n);
}
}
}.fp(Ref(n),fracs))
}
fractranW(2,fracs).walk(20).println();
Output:
L(2,15,825,725,1925,2275,425,390,330,290,770,910,170,156,132,116,308,364,68,4)

Translation of: Python
var [const] BN=Import("zklBigNum");  // libGMP
fcn fractranPrimes{
foreach n,fr in ([1..].zip(fractranW(BN(2),fracs))){
if(fr.num1s==1){
p:=(fr.toString(2) - "1").len();  // count zeros
if(p>1)
println("Prime %3d from the nth Fractran(%8d): %d".fmt(p,n,fr));
}
}
}
fractranPrimes();
Output:
Prime   2 from the nth Fractran(      20): 4
Prime   3 from the nth Fractran(      70): 8
Prime   5 from the nth Fractran(     281): 32
...
Prime 227 from the nth Fractran(15956646): 215679573337205118357336120696157045389097155380324579848828881993728
Prime 229 from the nth Fractran(16429799): 862718293348820473429344482784628181556388621521298319395315527974912
Prime 233 from the nth Fractran(17293373): 13803492693581127574869511724554050904902217944340773110325048447598592
Prime 239 from the nth Fractran(18633402): 883423532389192164791648750371459257913741948437809479060803100646309888
...
`