Coprime triplets: Difference between revisions
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# iterative Greatest Common Divisor routine, returns the gcd of m and n # |
# iterative Greatest Common Divisor routine, returns the gcd of m and n # |
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PROC gcd = ( INT m, n )INT: |
PROC gcd = ( INT m, n )INT: |
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BEGIN |
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n |
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ELSE |
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INT a := ABS m, b := ABS n; |
INT a := ABS m, b := ABS n; |
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WHILE b /= 0 DO |
WHILE b /= 0 DO |
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| Line 23: | Line 21: | ||
OD; |
OD; |
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a |
a |
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END # gcd # ; |
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# returns an array of the coprime triplets up to n # |
# returns an array of the coprime triplets up to n # |
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OP COPRIMETRIPLETS = ( INT n )[]INT: |
OP COPRIMETRIPLETS = ( INT n )[]INT: |
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Revision as of 16:10, 12 June 2021
- Task
Find and show the smallest number which is coprime to the last two predecessors and has not yet appeared; a(1)=1, a(2)=2.
p and q are coprimes if they have no common factors other than 1.
Let p, q < 50
ALGOL 68
<lang algol68>BEGIN # find members of the coprime triplets sequence: starting from 1, 2 the #
# subsequent members are the lowest number coprime to the previous two #
# that haven't appeared in the sequence yet #
# iterative Greatest Common Divisor routine, returns the gcd of m and n #
PROC gcd = ( INT m, n )INT:
BEGIN
INT a := ABS m, b := ABS n;
WHILE b /= 0 DO
INT new a = b;
b := a MOD b;
a := new a
OD;
a
END # gcd # ;
# returns an array of the coprime triplets up to n #
OP COPRIMETRIPLETS = ( INT n )[]INT:
BEGIN
[ 1 : n ]INT result;
IF n > 0 THEN
result[ 1 ] := 1;
IF n > 1 THEN
[ 1 : n ]BOOL used;
used[ 1 ] := used[ 2 ] := TRUE;
FOR i FROM 3 TO n DO used[ i ] := FALSE; result[ i ] := 0 OD;
result[ 2 ] := 2;
FOR i FROM 3 TO n DO
INT p1 = result[ i - 1 ];
INT p2 = result[ i - 2 ];
BOOL found := FALSE;
FOR j TO n WHILE NOT found DO
IF NOT used[ j ] THEN
found := gcd( p1, j ) = 1 AND gcd( p2, j ) = 1;
IF found THEN
used[ j ] := TRUE;
result[ i ] := j
FI
FI
OD
OD
FI
FI;
result
END # COPRIMETRIPLETS # ;
[]INT cps = COPRIMETRIPLETS 49;
INT printed := 0;
FOR i TO UPB cps DO
IF cps[ i ] /= 0 THEN
print( ( whole( cps[ i ], -3 ) ) );
printed +:= 1;
IF printed MOD 10 = 0 THEN print( ( newline ) ) FI
FI
OD;
print( ( newline, "Found ", whole( printed, 0 ), " coprime triplets up to ", whole( UPB cps, 0 ), newline ) )
END</lang>
- Output:
1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45 Found 36 coprime triplets up to 49
AppleScript
<lang applescript>on hcf(a, b)
repeat until (b = 0)
set x to a
set a to b
set b to x mod b
end repeat
if (a < 0) then return -a
return a
end hcf
on coprimeTriplets(max)
if (max < 3) then return {}
script o
property candidates : {}
property output : {1, 2}
end script
-- When repeatedly searching for lowest unused numbers, it's faster in
-- AppleScript to take numbers from a preset list of candidates which
-- grows shorter from at or near the low end as used numbers are removed
-- than it is to test increasing numbers of previous numbers each time
-- against a list that's growing longer with them.
-- Generate the list of candidates here.
repeat with i from 3 to max
set end of o's candidates to i
end repeat
set candidateCount to max - 2
set {p1, p2} to o's output
set ok to true
repeat while (ok) -- While suitable coprimes found and candidates left.
repeat with i from 1 to candidateCount
set q to item i of o's candidates
set ok to ((hcf(p1, q) is 1) and (hcf(p2, q) is 1))
if (ok) then -- q is coprime with both p1 and p2.
set end of o's output to q
set p1 to p2
set p2 to q
-- Remove q from the candidate list.
set item i of o's candidates to missing value
set o's candidates to o's candidates's numbers
set candidateCount to candidateCount - 1
set ok to (candidateCount > 0)
exit repeat
end if
end repeat
end repeat
return o's output
end coprimeTriplets
-- Task code: return coprimeTriplets(49)</lang>
- Output:
<lang applescript>{1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 6, 17, 19, 10, 21, 23, 16, 15, 29, 14, 25, 27, 22, 31, 35, 12, 37, 41, 18, 43, 47, 20, 33, 49, 26, 45}</lang>
Arturo
<lang rebol>lst: [1 2]
while [true][
n: 3 prev2: lst\[dec dec size lst] prev1: last lst
while -> any? @[
contains? lst n
1 <> gcd @[n prev2]
1 <> gcd @[n prev1]
] -> n: n + 1
if n >= 50 -> break 'lst ++ n
]
loop split.every:10 lst 'a ->
print map a => [pad to :string & 3]</lang>
- Output:
1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Delphi
<lang Delphi> program Coprime_triplets;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
//https://rosettacode.org/wiki/Greatest_common_divisor#Pascal_.2F_Delphi_.2F_Free_Pascal function Gcd(u, v: longint): longint; begin
if v = 0 then EXIT(u); result := Gcd(v, u mod v);
end;
function IsIn(value: Integer; a: TArray<Integer>): boolean; begin
for var e in a do
if e = value then
exit(true);
Result := false;
end;
function CoprimeTriplets(less_than: Integer = 50): TArray<Integer>; var
cpt: TArray<Integer>; _end: Integer;
begin
cpt := [1, 2]; _end := high(cpt);
while True do
begin
var m := 1;
while IsIn(m, cpt) or (gcd(m, cpt[_end]) <> 1) or (gcd(m, cpt[_end - 1]) <> 1) do
inc(m);
if m >= less_than then
exit(cpt);
SetLength(cpt, Length(cpt) + 1);
_end := high(cpt);
cpt[_end] := m;
end;
end;
begin
var trps := CoprimeTriplets();
writeln('Found ', length(trps), ' coprime triplets less than 50:');
for var i := 0 to High(trps) do
begin
write(trps[i]: 2, ' ');
if (i + 1) mod 10 = 0 then
writeln;
end;
{$IFNDEF UNIX} Readln; {$ENDIF}
end.</lang>
F#
<lang fsharp> // Coprime triplets: Nigel Galloway. May 12th., 2021 let rec fN g=function 0->g=1 |n->fN n (g%n) let rec fG t n1 n2=seq{let n=seq{1..0x0FFFFFFF}|>Seq.find(fun n->not(List.contains n t) && fN n1 n && fN n2 n) in yield n; yield! cT(n::t) n2 n} let cT=seq{yield 1; yield 2; yield! fG [1;2] 1 2} cT|>Seq.takeWhile((>)50)|>Seq.iter(printf "%d "); printfn "" </lang>
- Output:
1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Factor
<lang factor>USING: combinators.short-circuit.smart formatting grouping io kernel make math prettyprint sequences sets ;
- coprime? ( m n -- ? ) simple-gcd 1 = ;
- coprime-both? ( m n o -- ? ) '[ _ coprime? ] both? ;
- triplet? ( hs m n o -- ? )
{ [ coprime-both? nip ] [ 2nip swap in? not ] } && ;
- next ( hs m n -- hs' m' n' )
0 [ 4dup triplet? ] [ 1 + ] until nipd pick [ adjoin ] keepd ;
- (triplets-upto) ( n -- )
[ HS{ 1 2 } clone 1 , 1 2 ] dip
'[ 2dup [ _ < ] both? ] [ dup , next ] while 3drop ;
- triplets-upto ( n -- seq ) [ (triplets-upto) ] { } make ;
"Coprime triplets under 50:" print 50 triplets-upto [ 9 group simple-table. nl ] [ length "Found %d terms.\n" printf ] bi</lang>
- Output:
Coprime triplets under 50: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45 Found 36 terms.
FreeBASIC
<lang freebasic>function gcd( a as uinteger, b as uinteger ) as uinteger
if b = 0 then return a return gcd( b, a mod b )
end function
function num_in_array( array() as integer, num as integer ) as boolean
for i as uinteger = 1 to ubound(array)
if array(i) = num then return true
next i
return false
end function
redim as integer trips(1 to 2) trips(1) = 1 : trips(2) = 2 dim as integer last
do
last = ubound(trips)
for q as integer = 1 to 49
if not num_in_array( trips(), q ) _
andalso gcd(q, trips(last)) = 1 _
andalso gcd(q, trips(last-1)) = 1 then
redim preserve as integer trips( 1 to last+1 )
trips(last+1) = q
continue do
end if
next q
exit do
loop
print using "Found ## terms:"; ubound(trips)
for i as integer = 1 to last
print trips(i);" ";
next i : print</lang>
- Output:
Found 36 terms: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Go
<lang go>package main
import (
"fmt" "rcu"
)
func contains(a []int, v int) bool {
for _, e := range a {
if e == v {
return true
}
}
return false
}
func main() {
const limit = 50
cpt := []int{1, 2}
for {
m := 1
l := len(cpt)
for contains(cpt, m) || rcu.Gcd(m, cpt[l-1]) != 1 || rcu.Gcd(m, cpt[l-2]) != 1 {
m++
}
if m >= limit {
break
}
cpt = append(cpt, m)
}
fmt.Printf("Coprime triplets under %d:\n", limit)
for i, t := range cpt {
fmt.Printf("%2d ", t)
if (i+1)%10 == 0 {
fmt.Println()
}
}
fmt.Printf("\n\nFound %d such numbers\n", len(cpt))
}</lang>
- Output:
Coprime triplets under 50: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45 Found 36 such numbers
Haskell
<lang haskell>import Data.List (find, transpose, unfoldr) import Data.List.Split (chunksOf) import qualified Data.Set as S
COPRIME TRIPLES --------------------
coprimeTriples :: Integral a => [a] coprimeTriples =
[1, 2] <> unfoldr go (S.fromList [1, 2], (1, 2))
where
go (seen, (a, b)) =
Just
(c, (S.insert c seen, (b, c)))
where
Just c =
find
( ((&&) . flip S.notMember seen)
<*> ((&&) . coprime a <*> coprime b)
)
[3 ..]
coprime :: Integral a => a -> a -> Bool coprime a b = 1 == gcd a b
TEST -------------------------
main :: IO () main =
let xs = takeWhile (< 50) coprimeTriples
in putStrLn (show (length xs) <> " terms below 50:\n")
>> putStrLn
( spacedTable
justifyRight
(chunksOf 10 (show <$> xs))
)
FORMAT ------------------------
spacedTable ::
(Int -> Char -> String -> String) -> String -> String
spacedTable aligned rows =
unlines $
unwords
. zipWith
(`aligned` ' ')
(maximum . fmap length <$> transpose rows)
<$> rows
justifyRight :: Int -> Char -> String -> String justifyRight n c = (drop . length) <*> (replicate n c <>)</lang>
- Output:
36 terms below 50: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Julia
<lang julia>function coprime_triplets(less_than = 50)
cpt = [1, 2]
while true
m = 1
while m in cpt || gcd(m, cpt[end]) != 1 || gcd(m, cpt[end - 1]) != 1
m += 1
end
m >= less_than && return cpt
push!(cpt, m)
end
end
trps = coprime_triplets() println("Found $(length(trps)) coprime triplets less than 50:") foreach(p -> print(rpad(p[2], 3), p[1] %10 == 0 ? "\n" : ""), enumerate(trps))
</lang>
- Output:
Found 36 coprime triplets less than 50: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Nim
<lang Nim>import math, strutils
var list = @[1, 2]
while true:
var n = 3 let prev2 = list[^2] let prev1 = list[^1] while n in list or gcd(n, prev2) != 1 or gcd(n, prev1) != 1: inc n if n >= 50: break list.add n
echo list.join(" ")</lang>
- Output:
1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Perl
<lang perl>use strict; use warnings; use feature <state say>; use ntheory 'gcd'; use List::Util 'first'; use List::Lazy 'lazy_list'; use enum qw(False True); use constant Inf => 1e5;
my $ct = lazy_list {
state @c = (1, 2);
state %seen = (1 => True, 2 => True);
state $min = 3;
my $g = $c[-2] * $c[-1];
my $n = first { !$seen{$_} and gcd($_,$g) == 1 } $min .. Inf;
$seen{$n} = True;
$min = first { !$seen{$_} } $min .. Inf;
push @c, $n;
shift @c
};
my @ct; do { push @ct, $ct->next() } until $ct[-1] > 50; pop @ct; say join ' ', @ct</lang>
- Output:
1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Phix
function coprime_triplets(integer less_than=50) sequence cpt = {1,2} while true do integer m = 1 while find(m,cpt) or gcd(m,cpt[$])!=1 or gcd(m,cpt[$-1])!=1 do m += 1 end while if m>=less_than then exit end if cpt &= m end while return cpt end function sequence res = apply(true,sprintf,{{"%2d"},coprime_triplets()}) printf(1,"Found %d coprime triplets:\n%s\n",{length(res),join_by(res,1,10," ")})
- Output:
Found 36 coprime triplets: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45
Raku
<lang perl6>my @coprime-triplets = 1, 2, {
state %seen = 1, True, 2, True;
state $min = 3;
my $g = $^a * $^b;
my $n = ($min .. *).first: { !%seen{$_} && ($_ gcd $g == 1) }
%seen{$n} = True;
if %seen.elems %% 100 { $min = ($min .. *).first: { !%seen{$_} } }
$n
} … *;
put "Coprime triplets before first > 50:\n", @coprime-triplets[^(@coprime-triplets.first: * > 50, :k)].batch(10)».fmt("%4d").join: "\n";
put "\nOr maybe, minimum Coprime triplets that encompass 1 through 50:\n", @coprime-triplets[0..(@coprime-triplets.first: * == 42, :k)].batch(10)».fmt("%4d").join: "\n";
put "\nAnd for the heck of it: 1001st through 1050th Coprime triplet:\n", @coprime-triplets[1000..1049].batch(10)».fmt("%4d").join: "\n";</lang>
- Output:
Coprime triplets before first > 50: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45 Or maybe, minimum Coprime triplets that encompass 1 through 50: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45 53 28 39 55 32 51 59 38 61 63 34 65 57 44 67 69 40 71 73 24 77 79 30 83 89 36 85 91 46 75 97 52 81 95 56 87 101 50 93 103 58 99 107 62 105 109 64 111 113 68 115 117 74 119 121 48 125 127 42 And for the heck of it: 1001st through 1050th Coprime triplet: 682 1293 1361 680 1287 1363 686 1299 1367 688 1305 1369 692 1311 1373 694 1317 1375 698 1323 1381 704 1329 1379 706 1335 1387 716 1341 1385 712 1347 1391 700 1353 1399 710 1359 1393 718 1371 1397 722 1365 1403 724 1377 1405 728 1383
REXX
<lang rexx>/*REXX program finds and display coprime triplets below a specified limit (limit=50).*/ parse arg n cols . /*obtain optional arguments from the CL*/ if n== | n=="," then n= 50 /*Not specified? Then use the default.*/ if cols== | cols=="," then cols= 10 /* " " " " " " */ w= max(3, length( commas(n) ) ) /*width of a number in any column. */
@copt= ' coprime triplets where N < ' commas(n)
if cols>0 then say ' index │'center(@copt, 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(W+1), '─') !.= 0; @.= !.; idx= 1; $= /*initialize some variables. */
do #=1
do j=1; if @.j then iterate /*J in list of coprime triplets? Skip.*/
if #<3 then leave /*First two entries not defined? Use it*/
a= # - 1; b= # - 2 /*get the last two indices of sequence.*/
if gcd(j, !.a)\==1 then iterate /*J not coprime with last number?*/
if gcd(j, !.b)\==1 then iterate /*" " " " penultimate " */
leave /*OK, we've found a new coprime triplet*/
end /*j*/
if j>=n then leave /*Have we exceeded the limit? Then quit*/
@.j= 1; !.#= j /*flag a coprime triplet (two methods).*/
if cols==0 then iterate /*Not showing the numbers? Keep looking*/
$= $ right( commas(j), w) /*append coprime triplet to output list*/
if #//cols\==0 then iterate /*Is output line full? No, keep looking*/
say center(idx, 7)'│' substr($, 2); $= /*show output line of coprime triplets.*/
idx= idx + cols /*bump the index for the output line. */
end /*forever*/
if $\== then say center(idx, 7)'│' substr($, 2) /*show any residual output numbers*/ if cols>0 then say '───────┴'center("" , 1 + cols*(w+1), '─') say say 'Found ' commas(#-1) @copt 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 ? gcd: procedure; parse arg x,y; do until _==0; _= x//y; x= y; y= _; end; return x</lang>
- output when using the default inputs:
index │ coprime triplets where N < 50 ───────┼───────────────────────────────────────── 1 │ 1 2 3 5 4 7 9 8 11 13 11 │ 6 17 19 10 21 23 16 15 29 14 21 │ 25 27 22 31 35 12 37 41 18 43 31 │ 47 20 33 49 26 45 ───────┴───────────────────────────────────────── Found 36 coprime triplets where N < 50
Ring
<lang ring> see "working..." + nl row = 2 numbers = 1:50 first = 1 second = 2 see "Coprime triplets are:" + nl see "" + first + " " + second + " "
for n = 3 to len(numbers)
flag1 = 1
flag2 = 1
if first < numbers[n]
min = first
else
min = numbers[n]
ok
for m = 2 to min
if first%m = 0 and numbers[n]%m = 0
flag1 = 0
exit
ok
next
if second < numbers[n]
min = second
else
min = numbers[n]
ok
for m = 2 to min
if second%m = 0 and numbers[n]%m = 0
flag2 = 0
exit
ok
next
if flag1 = 1 and flag2 = 1
see "" + numbers[n] + " "
first = second
second = numbers[n]
del(numbers,n)
row = row+1
if row%10 = 0
see nl
ok
n = 2
ok
next
see nl + "Found " + row + " coprime triplets" + nl see "done..." + nl
</lang>
- Output:
working... Coprime triplets are: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45 Found 36 coprime triplets done...
Wren
<lang ecmascript>import "/math" for Int import "/seq" for Lst import "/fmt" for Fmt
var limit = 50 var cpt = [1, 2]
while (true) {
var m = 1
while (cpt.contains(m) || Int.gcd(m, cpt[-1]) != 1 || Int.gcd(m, cpt[-2]) != 1) {
m = m + 1
}
if (m >= limit) break
cpt.add(m)
} System.print("Coprime triplets under %(limit):") for (chunk in Lst.chunks(cpt, 10)) Fmt.print("$2d", chunk) System.print("\nFound %(cpt.count) such numbers.")</lang>
- Output:
Coprime triplets under 50: 1 2 3 5 4 7 9 8 11 13 6 17 19 10 21 23 16 15 29 14 25 27 22 31 35 12 37 41 18 43 47 20 33 49 26 45 Found 36 such numbers.