Initial data commit

Task/Goldbachs-comet/00-META.yaml

@@ -0,0 +1,3 @@
+---
+from: http://rosettacode.org/wiki/Goldbach's_comet
+note: Prime Numbers

Task/Goldbachs-comet/00-TASK.txt

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+'''Goldbach's comet''' is the name given to a plot of the function '''g(E)''', the so-called '''Goldbach function'''.
+
+The Goldbach function is studied in relation to Goldbach's conjecture. The function '''g(E)''' is defined for all ''even'' integers '''E>2''' to be the number of different ways in which E can be expressed as the sum of two primes. 
+
+;Examples
+
+* G(4) = 1, since 4 can only be expressed as the sum of one distinct pair of primes (4 = 2 + 2)
+* G(22) = 3, since 22 can be expressed as the sum of 3 distinct pairs of primes (22 = 11 + 11 = 5 + 17 = 3 + 19)
+
+
+;Task
+
+* Find and show (preferably, in a neat 10x10 table) the first 100 G numbers (that is: the result of the G function described above, for the first 100 even numbers >= 4)
+* Find and display the value of G(1000000)
+
+
+;Stretch
+* Calculate the values of G up to 2000 (inclusive) and display the results in a scatter 2d-chart, aka the [https://upload.wikimedia.org/wikipedia/en/f/fb/Goldbachs_comet.gif Goldbach's Comet]
+
+
+;See also
+;* [[wp:Goldbach's conjecture|Wikipedia: Goldbach's conjecture]]
+;* [[oeis:A045917|OEIS: A045917 - From Goldbach problem: number of decompositions of 2n into unordered sums of two primes]]
+
+
+
+

Task/Goldbachs-comet/11l/goldbachs-comet.11l

@@ -0,0 +1,26 @@
+F is_prime(a)
+   I a == 2
+      R 1B
+   I a < 2 | a % 2 == 0
+      R 0B
+   L(i) (3 .. Int(sqrt(a))).step(2)
+      I a % i == 0
+         R 0B
+   R 1B
+
+F g(n)
+   assert(n > 2 & n % 2 == 0, ‘n in goldbach function g(n) must be even’)
+   V count = 0
+   L(i) 1 .. n I/ 2
+      I is_prime(i) & is_prime(n - i)
+         count++
+   R count
+
+print(‘The first 100 G numbers are:’)
+
+V col = 1
+L(n) (4.<204).step(2)
+   print(String(g(n)).ljust(4), end' I (col % 10 == 0) {"\n"} E ‘’)
+   col++
+
+print("\nThe value of G(1000000) is "g(1'000'000))

Task/Goldbachs-comet/ALGOL-68/goldbachs-comet.alg

@@ -0,0 +1,63 @@
+BEGIN # calculate values of the Goldbach function G where G(n) is the number #
+      # of prime pairs that sum to n, n even and > 2                         #
+      # generates an ASCII scatter plot of G(n) up to G(2000)                #
+      # (Goldbach's Comet)                                                   #
+    PR read "primes.incl.a68" PR          # include prime utilities          #
+    INT max prime = 1 000 000;            # maximum number we will consider  #
+    INT max plot  =     2 000;            # maximum G value for the comet    #
+    []BOOL prime = PRIMESIEVE max prime;  # sieve of primes to max prime     #
+    [ 0 : max plot ]INT g2;               # table of G values: g2[n] = G(2n) #
+    # construct the table of G values #
+    FOR n FROM LWB g2 TO UPB g2 DO g2[ n ] := 0 OD;
+    g2[ 4 ] := 1;                     # 4 is the only sum of two even primes #
+    FOR p FROM 3 BY 2 TO max plot OVER 2 DO
+        IF prime[ p ] THEN
+            g2[ p + p ] +:= 1;
+            FOR q FROM p + 2 BY 2 TO max plot - p DO
+                IF prime[ q ] THEN
+                    g2[ p + q ] +:= 1
+                FI
+            OD
+        FI
+    OD;
+    # show the first hundred G values #
+    INT c := 0;
+    FOR n FROM 4 BY 2 TO 202 DO
+        print( ( whole( g2[ n ], -4 ) ) );
+        IF ( c +:= 1 ) = 10 THEN print( ( newline ) ); c := 0 FI
+    OD;
+    # show G( 1 000 000 ) #
+    INT gm := 0;
+    FOR p FROM 3 TO max prime OVER 2 DO
+        IF prime[ p ] THEN
+            IF prime[ max prime - p ] THEN
+                gm +:= 1
+            FI
+        FI
+    OD;
+    print( ( "G(", whole( max prime, 0 ), "): ", whole( gm, 0 ), newline ) );
+    # find the maximum value of G up to the maximum plot size #
+    INT max g := 0;
+    FOR n FROM 2 BY 2 TO max plot DO
+        IF g2[ n ] > max g THEN max g := g2[ n ] FI
+    OD;
+    # draw an ASCII scatter plot of G, each position represents 5 G values #
+    # the vertical axis is n/10, the horizontal axis is G(n) #
+    INT plot step = 10;
+    STRING plot value = " .-+=*%$&#@";
+    FOR g FROM 0 BY plot step TO max plot - plot step DO
+        [ 0 : max g ]INT values;
+        FOR v pos FROM LWB values TO UPB values DO values[ v pos ] := 0 OD;
+        INT max v := 0;
+        FOR g element FROM g BY 2 TO g + ( plot step - 1 ) DO
+            INT g2 value = g2[ g element ];
+            values[ g2 value ] +:= 1;
+            IF g2 value > max v THEN max v := g2 value FI
+        OD;
+        print( ( IF g MOD 100 = 90 THEN "+" ELSE "|" FI ) );
+        FOR v pos FROM 1 TO max v DO # exclude 0 values from the plot #
+            print( ( plot value[ values[ v pos ] + 1 ] ) )
+        OD;
+        print( ( newline ) )
+    OD
+END

Task/Goldbachs-comet/AWK/goldbachs-comet.awk

@@ -0,0 +1,37 @@
+# syntax: GAWK -f GOLDBACHS_COMET.AWK
+BEGIN {
+    print("The first 100 G numbers:")
+    for (n=4; n<=202; n+=2) {
+      printf("%4d%1s",g(n),++count%10?"":"\n")
+    }
+    n = 1000000
+    printf("\nG(%d): %d\n",n,g(n))
+    n = 4
+    printf("G(%d): %d\n",n,g(n))
+    n = 22
+    printf("G(%d): %d\n",n,g(n))
+    exit(0)
+}
+function g(n,  count,i) {
+    if (n % 2 == 0) { # n must be even
+      for (i=2; i<=(1/2)*n; i++) {
+        if (is_prime(i) && is_prime(n-i)) {
+          count++
+        }
+      }
+    }
+    return(count)
+}
+function is_prime(n,  d) {
+    d = 5
+    if (n < 2) { return(0) }
+    if (n % 2 == 0) { return(n == 2) }
+    if (n % 3 == 0) { return(n == 3) }
+    while (d*d <= n) {
+      if (n % d == 0) { return(0) }
+      d += 2
+      if (n % d == 0) { return(0) }
+      d += 4
+    }
+    return(1)
+}

Task/Goldbachs-comet/Arturo/goldbachs-comet.arturo

@@ -0,0 +1,18 @@
+G: function [n][
+    size select 2..n/2 'x ->
+        and? [prime? x][prime? n-x]
+]
+
+print "The first 100 G values:"
+loop split.every: 10 map select 4..202 => even? => G 'row [
+    print map to [:string] row 'item -> pad item 3
+]
+
+print ["\nG(1000000) =" G 1000000]
+
+csv: join.with:",\n" map select 4..2000 => even? 'x ->
+    ~"|x|, |G x|"
+
+; write the CSV data to a file which we can then visualize
+; via our preferred spreadsheet app
+write "comet.csv" csv

Task/Goldbachs-comet/AutoHotkey/goldbachs-comet.ahk

@@ -0,0 +1,22 @@
+c := 0
+while (c<100)
+    if (x := g(A_Index))
+        c++, result .= x (Mod(c, 10) ? "`t" : "`n")
+MsgBox % result "`ng(1000000) : " g(1000000)
+return
+
+g(n, i:=1) {
+    if Mod(n, 2)
+        return false
+    while (++i <= n/2)
+        if (is_prime(i) && is_prime(n-i))
+            count++
+    return count
+}
+
+is_prime(N) {
+    Loop, % Floor(Sqrt(N))
+        if (A_Index > 1 && !Mod(N, A_Index))
+            Return false
+    Return true
+}

Task/Goldbachs-comet/Delphi/goldbachs-comet.delphi

@@ -0,0 +1,31 @@
+function GetGoldbachCount(N: integer): integer;
+{Count number of prime number combinations add up to N }
+var I: integer;
+begin
+Result:=0;
+{Look at all number pairs that add up to N}
+{And see if they are prime}
+for I:=1 to N div 2 do
+ if IsPrime(I) and IsPrime(N-I) then Inc(Result);
+end;
+
+procedure ShowGoldbachComet(Memo: TMemo);
+{Show first 100 Goldback numbers}
+var I,N,Cnt,C: integer;
+var S: string;
+begin
+Cnt:=0; N:=2; S:='';
+while true do
+	begin
+	C:=GetGoldbachCount(N);
+	if C>0 then
+		begin
+		Inc(Cnt);
+		S:=S+Format('%3d',[C]);
+		if (Cnt mod 10)=0 then S:=S+CRLF;
+		if Cnt>=100 then break;
+		end;
+	Inc(N,2);
+	end;
+Memo.Lines.Add(S);
+end;

Task/Goldbachs-comet/EasyLang/goldbachs-comet.easy

@@ -0,0 +1,40 @@
+proc isprime n . r .
+   r = 1
+   if n <= 1
+      r = 0
+      break 1
+   .
+   if n mod 2 = 0
+      if n = 2
+         break 1
+      .
+      r = 0
+      break 1
+   .
+   for i = 3 step 2 to sqrt n
+      if n mod i = 0
+         r = 0
+         break 2
+      .
+   .
+.
+proc goldbach n . cnt .
+   cnt = 0
+   for i = 1 to n div 2
+      call isprime i r
+      if r = 1
+         call isprime n - i r
+         cnt += r
+      .
+   .
+.
+numfmt 0 3
+for n = 4 step 2 to 202
+   call goldbach n r
+   write r
+   if n mod 20 = 2
+      print ""
+   .
+.
+call goldbach 1000000 r
+print r

Task/Goldbachs-comet/FreeBASIC/goldbachs-comet-1.basic

@@ -0,0 +1,29 @@
+Function isPrime(Byval ValorEval As Uinteger) As Boolean
+    If ValorEval <= 1 Then Return False
+    For i As Integer = 2 To Int(Sqr(ValorEval))
+        If ValorEval Mod i = 0 Then Return False
+    Next i
+    Return True
+End Function
+
+Function g(n As Uinteger) As Uinteger
+    Dim As Uinteger i, count = 0
+    If (n Mod 2 = 0) Then     'n in goldbach function g(n) must be even
+        For i = 2 To (1/2) * n
+            If isPrime(i) And isPrime(n - i) Then count += 1
+        Next i
+    End If
+    Return count
+End Function
+
+Print "The first 100 G numbers are:"
+
+Dim As Uinteger col = 1
+For n As Uinteger = 4 To 202 Step 2
+    Print Using "####"; g(n);
+    If (col Mod 10 = 0) Then Print
+    col += 1
+Next n
+
+Print !"\nThe value of G(1000000) is "; g(1000000)
+Sleep

Task/Goldbachs-comet/FreeBASIC/goldbachs-comet-2.basic

@@ -0,0 +1,47 @@
+OPTION STRICT: OPTION DEFINT
+VAR MAX_G = 4000, MAX_P = 1000000
+VAR ROOT_MAX_P = FLOOR(SQR(MAX_P))
+VAR HALF_MAX_G = MAX_G DIV 2
+VAR G[MAX_G + 1], P[MAX_P + 1]
+VAR I, J
+CLS: GCLS
+P[0] = FALSE: P[1] = 0: P[2] = TRUE
+FOR I = 4 TO MAX_P STEP 2 P[I] = FALSE: NEXT
+FOR I = 3 TO MAX_P STEP 2 P[I] = TRUE: NEXT
+FOR I = 3 TO ROOT_MAX_P STEP 2
+  IF P[I] THEN
+    FOR J = I * I TO MAX_P STEP I
+      P[J] = FALSE
+    NEXT
+  ENDIF
+NEXT
+FOR I = 1 TO MAX_G G[I] = 0: NEXT
+G[4] = 1 ' 4 is the only sum of 2 even primes
+FOR I = 3 TO HALF_MAX_G STEP 2
+  IF P[I] THEN
+    INC G[I + 1]
+    FOR J = I + 2 TO MAX_G - 1
+      IF P[J] THEN
+        INC G[I + 1]
+      ENDIF
+    NEXT
+  ENDIF
+NEXT
+VAR C = 0
+FOR I = 4 TO 202 STEP 2
+  PRINT FORMAT$("%3D", G[I]),
+  INC C
+  IF C == 10 THEN PRINT: C = 0: ENDIF
+NEXT
+VAR GM = 0
+FOR I = 3 TO MAX_P DIV 2 STEP 2
+  IF P[I] THEN
+    IF P[MAX_P - I] THEN INC GM: ENDIF
+  ENDIF
+NEXT
+PRINT FORMAT$("G(%D): ", MAX_P); GM
+FOR I = 2 TO MAX_G - 10 STEP 10
+  FOR J = 1 TO I + 8 STEP 2
+    GPSET I DIV 10, 240-G[J],RGB(255,255,255)
+  NEXT
+NEXT

Task/Goldbachs-comet/J/goldbachs-comet-1.j

@@ -0,0 +1,11 @@
+   10 10$#/.~4,/:~ 0-.~,(<:/~ * +/~) p:1+i.p:inv 202
+ 1  1  1  2  1  2  2  2  2  3
+ 3  3  2  3  2  4  4  2  3  4
+ 3  4  5  4  3  5  3  4  6  3
+ 5  6  2  5  6  5  5  7  4  5
+ 8  5  4  9  4  5  7  3  6  8
+ 5  6  8  6  7 10  6  6 12  4
+ 5 10  3  7  9  6  5  8  7  8
+11  6  5 12  4  8 11  5  8 10
+ 5  6 13  9  6 11  7  7 14  6
+ 8 13  5  8 11  7  9 13  8  9

Task/Goldbachs-comet/J/goldbachs-comet-2.j

@@ -0,0 +1,2 @@
+   -:+/1 p: 1e6-p:i.p:inv 1e6
+5402

Task/Goldbachs-comet/Jq/goldbachs-comet-1.jq

@@ -0,0 +1,25 @@
+def count(s): reduce s as $_ (0; .+1);
+
+def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
+
+def nwise($n):
+  def n: if length <= $n then . else .[0:$n] , (.[$n:] | n) end;
+  n;
+
+def is_prime:
+  . as $n
+  | if ($n < 2)         then false
+    elif ($n % 2 == 0)  then $n == 2
+    elif ($n % 3 == 0)  then $n == 3
+    elif ($n % 5 == 0)  then $n == 5
+    elif ($n % 7 == 0)  then $n == 7
+    elif ($n % 11 == 0) then $n == 11
+    elif ($n % 13 == 0) then $n == 13
+    elif ($n % 17 == 0) then $n == 17
+    elif ($n % 19 == 0) then $n == 19
+    else
+      ($n | sqrt) as $rt
+      | 23
+      | until( . > $rt or ($n % . == 0); .+2)
+      | . > $rt
+    end;

Task/Goldbachs-comet/Jq/goldbachs-comet-2.jq

@@ -0,0 +1,15 @@
+# emit nothing if . is odd
+def G:
+  select(. % 2 == 0)
+  | count( range(2; (./2)+1) as $i
+           | select(($i|is_prime) and ((.-$i)|is_prime)) );
+
+def task1:
+  "The first 100 G numbers:",
+  ([range(4; 203; 2) | G] | nwise(10) | map(lpad(4)) | join(" "));
+
+def task($n):
+  $n, 4, 22
+  |"G(\(.)): \(G)";
+
+task1, "", task(1000000)

Task/Goldbachs-comet/Julia/goldbachs-comet.julia

@@ -0,0 +1,15 @@
+using Combinatorics
+using Plots
+using Primes
+
+g(n) = iseven(n) ? count(p -> all(isprime, p), partitions(n, 2)) : error("n must be even")
+
+println("The first 100 G numbers are: ")
+
+foreach(p -> print(lpad(p[2], 4), p[1] % 10 == 0 ? "\n" : ""), map(g, 4:2:202) |> enumerate)
+
+println("\nThe value of G(1000000) is ", g(1_000_000))
+
+x = collect(2:2002)
+y = map(g, 2x)
+scatter(x, y, markerstrokewidth = 0, color = ["red", "blue", "green"][mod1.(x, 3)])

Task/Goldbachs-comet/Lua/goldbachs-comet.lua

@@ -0,0 +1,29 @@
+function T(t) return setmetatable(t, {__index=table}) end
+table.range = function(t,n) local s=T{} for i=1,n do s[i]=i end return s end
+table.map = function(t,f) local s=T{} for i=1,#t do s[i]=f(t[i]) end return s end
+table.batch = function(t,n,f) for i=1,#t,n do local s=T{} for j=1,n do s[j]=t[i+j-1] end f(s) end return t end
+
+function isprime(n)
+  if n < 2 then return false end
+  if n % 2 == 0 then return n==2 end
+  if n % 3 == 0 then return n==3 end
+  for f = 5, n^0.5, 6 do
+    if n%f==0 or n%(f+2)==0 then return false end
+  end
+  return true
+end
+
+function goldbach(n)
+  local count = 0
+  for i = 1, n/2 do
+    if isprime(i) and isprime(n-i) then
+      count = count + 1
+    end
+  end
+  return count
+end
+
+print("The first 100 G numbers:")
+g = T{}:range(100):map(function(n) return goldbach(2+n*2) end)
+g:map(function(n) return string.format("%2d ",n) end):batch(10,function(t) print(t:concat()) end)
+print("G(1000000) = "..goldbach(1000000))

Task/Goldbachs-comet/Mathematica/goldbachs-comet.math

@@ -0,0 +1,8 @@
+ClearAll[GoldbachFuncion]
+GoldbachFuncion[e_Integer] := Module[{ps},
+  ps = Prime[Range[PrimePi[e/2]]];
+  Total[Boole[PrimeQ[e - ps]]]
+]
+Grid[Partition[GoldbachFuncion /@ Range[4, 220, 2], 10]]
+GoldbachFuncion[10^6]
+DiscretePlot[GoldbachFuncion[e], {e, 4, 2000}, Filling -> None]

Task/Goldbachs-comet/Nim/goldbachs-comet.nim

@@ -0,0 +1,46 @@
+import std/[math, strformat, strutils, sugar]
+import chroma, plotly
+
+const
+  N1 = 100        # For part 1 of task.
+  N2 = 1_000_000  # For part 2 of task.
+  N3 = 2000       # For stretch part.
+
+# Erathostenes sieve.
+var isPrime: array[1..N1, bool]
+for i in 2..N1: isPrime[i] = true
+for n in 2..sqrt(N1.toFloat).int:
+  for k in countup(n * n, N1, n):
+    isPrime[k] = false
+
+proc g(n: int): int =
+  ## Goldbach function.
+  assert n > 2 and n mod 2 == 0, "“n” must be even and greater than 2."
+  for i in 1..(n div 2):
+    if isPrime[i] and isPrime[n - i]:
+      inc result
+
+# Part 1.
+echo &"First {N1} G numbers:"
+var col = 1
+for n in 2..N1:
+  stdout.write align($g( 2 * n), 3)
+  stdout.write if col mod 10 == 0: '\n' else: ' '
+  inc col
+
+# Part 2.
+echo &"\nG({N2}) = ", g(N2)
+
+# Stretch part.
+
+const Colors = collect(for name in ["red", "blue", "green"]: name.parseHtmlName())
+var x, y: seq[float]
+var colors: seq[Color]
+for n in 2..N3:
+  x.add n.toFloat
+  y.add g(2 * n).toFloat
+  colors.add Colors[n mod 3]
+
+let trace = Trace[float](type: Scatter, mode: Markers, marker: Marker[float](color: colors), xs: x, ys: y)
+let layout = Layout(title: "Goldbach’s comet", width: 1200, height: 400)
+Plot[float64](layout: layout, traces: @[trace]).show(removeTempFile = true)

Task/Goldbachs-comet/Perl/goldbachs-comet.pl

@@ -0,0 +1,36 @@
+use strict;
+use warnings;
+use feature 'say';
+
+use List::Util 'max';
+use GD::Graph::bars;
+use ntheory 'is_prime';
+
+sub table { my $t = shift() * (my $c = 1 + max map {length} @_); ( sprintf( ('%'.$c.'s')x@_, @_) ) =~ s/.{1,$t}\K/\n/gr }
+
+sub G {
+    my($n) = @_;
+    scalar grep { is_prime($_) and is_prime($n - $_) } 2 .. $n/2;
+}
+
+my @y;
+push @y, G(2*$_ + 4) for my @x = 0..1999;
+
+say $_ for table 10, @y;
+printf "G $_: %d", G($_) for 1e6;
+
+my @data = ( \@x, \@y);
+my $graph = GD::Graph::bars->new(1200, 400);
+$graph->set(
+    title          => q/Goldbach's Comet/,
+    y_max_value    => 170,
+    x_tick_number  => 10,
+    r_margin       => 10,
+    dclrs          => [ 'blue' ],
+) or die $graph->error;
+my $gd = $graph->plot(\@data) or die $graph->error;
+
+open my $fh, '>', 'goldbachs-comet.png';
+binmode $fh;
+print $fh $gd->png();
+close $fh;

Task/Goldbachs-comet/Phix/goldbachs-comet.phix

@@ -0,0 +1,49 @@
+(phixonline)-->
+ <span style="color: #000080;font-style:italic;">--
+ -- demo\rosetta\Goldbachs_comet.exw
+ -- ================================
+ --
+ -- Note: this plots n/2 vs G(n) for n=6 to 4000 by 2, matching wp and
+ --       Algol 68, Python, and Raku. However, while not wrong, Arturo
+ --       and Wren apparently plot n vs G(n) for n=6 to 2000 by 2, so
+ --       should you spot any (very) minor differences, that'd be why.
+ --</span>
+ <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
+ <span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.2"</span><span style="color: #0000FF;">)</span>
+
+ <span style="color: #008080;">constant</span> <span style="color: #000000;">limit</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">4000</span>
+ <span style="color: #004080;">sequence</span> <span style="color: #000000;">primes</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">get_primes_le</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">)[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$],</span>
+        <span style="color: #000000;">goldbach</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">reinstate</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">),{</span><span style="color: #000000;">4</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">})</span>
+ <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">primes</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
+     <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">i</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">primes</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
+         <span style="color: #004080;">integer</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">primes</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">primes</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
+         <span style="color: #008080;">if</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">></span><span style="color: #000000;">limit</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
+         <span style="color: #000000;">goldbach</span><span style="color: #0000FF;">[</span><span style="color: #000000;">s</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
+     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
+ <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
+
+ <span style="color: #004080;">sequence</span> <span style="color: #000000;">fhg</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">extract</span><span style="color: #0000FF;">(</span><span style="color: #000000;">goldbach</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">tagstart</span><span style="color: #0000FF;">(</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">100</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">))</span>
+ <span style="color: #004080;">string</span> <span style="color: #000000;">fhgs</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">join_by</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fhg</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%2d"</span><span style="color: #0000FF;">)</span>
+ <span style="color: #004080;">integer</span> <span style="color: #000000;">gm</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">apply</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1e6</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">get_primes_le</span><span style="color: #0000FF;">(</span><span style="color: #000000;">499999</span><span style="color: #0000FF;">)[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]),</span><span style="color: #7060A8;">is_prime</span><span style="color: #0000FF;">))</span>
+ <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"The first 100 G values:\n%s\n\nG(1,000,000) = %,d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">fhgs</span><span style="color: #0000FF;">,</span><span style="color: #000000;">gm</span><span style="color: #0000FF;">})</span>
+
+ <span style="color: #008080;">include</span> <span style="color: #000000;">pGUI</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
+ <span style="color: #008080;">include</span> <span style="color: #000000;">IupGraph</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
+
+ <span style="color: #008080;">function</span> <span style="color: #000000;">get_data</span><span style="color: #0000FF;">(</span><span style="color: #004080;">Ihandle</span> <span style="color: #000080;font-style:italic;">/*graph*/</span><span style="color: #0000FF;">)</span>
+     <span style="color: #008080;">return</span> <span style="color: #0000FF;">{{</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">),</span><span style="color: #7060A8;">extract</span><span style="color: #0000FF;">(</span><span style="color: #000000;">goldbach</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">)),</span><span style="color: #004600;">CD_RED</span><span style="color: #0000FF;">},</span>
+             <span style="color: #0000FF;">{</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">),</span><span style="color: #7060A8;">extract</span><span style="color: #0000FF;">(</span><span style="color: #000000;">goldbach</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">8</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">)),</span><span style="color: #004600;">CD_BLUE</span><span style="color: #0000FF;">},</span>
+             <span style="color: #0000FF;">{</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">),</span><span style="color: #7060A8;">extract</span><span style="color: #0000FF;">(</span><span style="color: #000000;">goldbach</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">)),</span><span style="color: #004600;">CD_DARK_GREEN</span><span style="color: #0000FF;">}}</span>
+ <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
+
+ <span style="color: #7060A8;">IupOpen</span><span style="color: #0000FF;">()</span>
+ <span style="color: #004080;">Ihandle</span> <span style="color: #000000;">graph</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">IupGraph</span><span style="color: #0000FF;">(</span><span style="color: #000000;">get_data</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"RASTERSIZE=640x440,MARKSTYLE=PLUS"</span><span style="color: #0000FF;">)</span>
+ <span style="color: #7060A8;">IupSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"XTICK=%d,XMIN=0,XMAX=%d"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">/</span><span style="color: #000000;">20</span><span style="color: #0000FF;">,</span><span style="color: #000000;">limit</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">})</span>
+ <span style="color: #7060A8;">IupSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"YTICK=20,YMIN=0,YMAX=%d"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">max</span><span style="color: #0000FF;">(</span><span style="color: #000000;">goldbach</span><span style="color: #0000FF;">)+</span><span style="color: #000000;">20</span><span style="color: #0000FF;">})</span>
+ <span style="color: #004080;">Ihandle</span> <span style="color: #000000;">dlg</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">IupDialog</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">`TITLE="Goldbach's comet",MINSIZE=400x300`</span><span style="color: #0000FF;">)</span>
+ <span style="color: #7060A8;">IupShow</span><span style="color: #0000FF;">(</span><span style="color: #000000;">dlg</span><span style="color: #0000FF;">)</span>
+ <span style="color: #008080;">if</span> <span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()!=</span><span style="color: #004600;">JS</span> <span style="color: #008080;">then</span>
+     <span style="color: #7060A8;">IupMainLoop</span><span style="color: #0000FF;">()</span>
+     <span style="color: #7060A8;">IupClose</span><span style="color: #0000FF;">()</span>
+ <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
+<!--

Task/Goldbachs-comet/Picat/goldbachs-comet.picat

@@ -0,0 +1,12 @@
+main =>
+  println("First 100 G numbers:"),
+  foreach({G,I} in zip(take([G: T in 1..300, G=g(T),G>0],100),1..100))
+    printf("%2d %s",G,cond(I mod 10 == 0,"\n",""))
+  end,
+  nl,
+  printf("G(1_000_000): %d\n", g(1_000_000)).
+
+g(N) = cond((N > 2, N mod 2 == 0),
+              {1 : I in 1..N // 2,
+                 prime(I),prime(N-I)}.len,
+              0).

Task/Goldbachs-comet/Python/goldbachs-comet.py

@@ -0,0 +1,25 @@
+from matplotlib.pyplot import scatter, show
+from sympy import isprime
+
+def g(n):
+    assert n > 2 and n % 2 == 0, 'n in goldbach function g(n) must be even'
+    count = 0
+    for i in range(1, n//2 + 1):
+        if isprime(i) and isprime(n - i):
+            count += 1
+    return count
+
+print('The first 100 G numbers are:')
+
+col = 1
+for n in range(4, 204, 2):
+    print(str(g(n)).ljust(4), end = '\n' if (col % 10 == 0) else '')
+    col += 1
+
+print('\nThe value of G(1000000) is', g(1_000_000))
+
+x = range(4, 4002, 2)
+y = [g(i) for i in x]
+colors = [["red", "blue", "green"][(i // 2) % 3] for i in x]
+scatter([i // 2 for i in x], y, marker='.', color = colors)
+show()

Task/Goldbachs-comet/Raku/goldbachs-comet.raku

@@ -0,0 +1,22 @@
+sub G (Int $n) { +(2..$n/2).grep: { .is-prime && ($n - $_).is-prime } }
+
+# Task
+put "The first 100 G values:\n", (^100).map({ G 2 × $_ + 4 }).batch(10)».fmt("%2d").join: "\n";
+
+put "\nG 1_000_000 = ", G 1_000_000;
+
+# Stretch
+use SVG;
+use SVG::Plot;
+
+my @x = map 2 × * + 4, ^2000;
+my @y = @x.map: &G;
+
+'Goldbachs-Comet-Raku.svg'.IO.spurt: SVG.serialize: SVG::Plot.new(
+    width       => 1000,
+    height      => 500,
+    background  => 'white',
+    title       => "Goldbach's Comet",
+    x           => @x,
+    values      => [@y,],
+).plot: :points;

Task/Goldbachs-comet/Ruby/goldbachs-comet.rb

@@ -0,0 +1,10 @@
+require 'prime'
+
+n = 100
+puts "The first #{n} Godbach numbers are: "
+sums = Prime.each(n*2 + 2).to_a[1..].repeated_combination(2).map(&:sum)
+sums << 4
+sums.sort.tally.values[...n].each_slice(10){|slice| puts "%4d"*slice.size % slice}
+
+n = 1000000
+puts "\nThe value of G(#{n}) is #{Prime.each(n/2).count{|pr| (n-pr).prime?} }."

Task/Goldbachs-comet/Rust/goldbachs-comet.rust

@@ -0,0 +1,75 @@
+// [dependencies]
+// primal = "0.3"
+// plotters = "0.3.2"
+
+use plotters::prelude::*;
+
+fn goldbach(n: u64) -> u64 {
+    let mut p = 2;
+    let mut count = 0;
+    loop {
+        let q = n - p;
+        if q < p {
+            break;
+        }
+        if primal::is_prime(p) && primal::is_prime(q) {
+            count += 1;
+        }
+        if p == 2 {
+            p += 1;
+        } else {
+            p += 2;
+        }
+    }
+    count
+}
+
+fn goldbach_plot(filename: &str) -> Result<(), Box<dyn std::error::Error>> {
+    let gvalues : Vec<u64> = (1..=2000).map(|x| goldbach(2 * x + 2)).collect();
+    let mut gmax = *gvalues.iter().max().unwrap();
+    gmax = 10 * ((gmax + 9) / 10);
+
+    let root = SVGBackend::new(filename, (1000, 500)).into_drawing_area();
+    root.fill(&WHITE)?;
+
+    let mut chart = ChartBuilder::on(&root)
+        .x_label_area_size(20)
+        .y_label_area_size(20)
+        .margin(10)
+        .caption("Goldbach's Comet", ("sans-serif", 24).into_font())
+        .build_cartesian_2d(0usize..2000usize, 0u64..gmax)?;
+
+    chart
+        .configure_mesh()
+        .disable_x_mesh()
+        .disable_y_mesh()
+        .draw()?;
+
+    chart.draw_series(
+        gvalues
+            .iter()
+            .cloned()
+            .enumerate()
+            .map(|p| Circle::new(p, 2, BLUE.filled())),
+    )?;
+
+    Ok(())
+}
+
+fn main() {
+    println!("First 100 G numbers:");
+    for i in 1..=100 {
+        print!(
+            "{:2}{}",
+            goldbach(2 * i + 2),
+            if i % 10 == 0 { "\n" } else { " " }
+        );
+    }
+
+    println!("\nG(1000000) = {}", goldbach(1000000));
+
+    match goldbach_plot("goldbach.svg") {
+        Ok(()) => {}
+        Err(error) => eprintln!("Error: {}", error),
+    }
+}

Task/Goldbachs-comet/Wren/goldbachs-comet.wren

@@ -0,0 +1,78 @@
+import "dome" for Window
+import "graphics" for Canvas, Color
+import "./math2" for Int
+import "./iterate" for Stepped
+import "./fmt" for Fmt
+import "./plot" for Axes
+
+var limit = 4002
+var primes = Int.primeSieve(limit-1).skip(1).toList
+var goldbach = {4: 1}
+for (i in Stepped.new(6..limit, 2)) goldbach[i] = 0
+for (i in 0...primes.count) {
+    for (j in i...primes.count) {
+        var s = primes[i] + primes[j]
+        if (s > limit) break
+        goldbach[s] = goldbach[s] + 1
+    }
+}
+
+System.print("The first 100 G values:")
+var count = 0
+for (i in Stepped.new(4..202, 2)) {
+    count = count + 1
+    Fmt.write("$2d ", goldbach[i])
+    if (count % 10 == 0) System.print()
+}
+
+primes = Int.primeSieve(499999).skip(1)
+var gm = 0
+for (p in primes) {
+    if (Int.isPrime(1e6 - p)) gm = gm + 1
+}
+System.print("\nG(1000000) = %(gm)")
+
+var Red   = []
+var Blue  = []
+var Green = []
+
+// create lists for the first 2000 G values for plotting by DOME.
+for(e in Stepped.new(4..limit, 2)) {
+    var c = e % 6
+    var n = e/2 - 1
+    if (c == 0) {
+        Red.add([n, goldbach[e]])
+    } else if (c == 2) {
+        Blue.add([n, goldbach[e]])
+    } else {
+        Green.add([n, goldbach[e]])
+    }
+}
+
+class Main {
+    construct new() {
+        Window.title = "Goldbach's comet"
+        Canvas.resize(1000, 600)
+        Window.resize(1000, 600)
+        Canvas.cls(Color.white)
+        var axes = Axes.new(100, 500, 800, 400, 0..2000, 0..200)
+        axes.draw(Color.black, 2)
+        var xMarks = Stepped.new(0..2000, 200)
+        var yMarks = Stepped.new(0..200, 20)
+        axes.mark(xMarks, yMarks, Color.black, 2)
+        var xMarks2 = Stepped.new(0..2000, 400)
+        var yMarks2 = Stepped.new(0..200, 40)
+        axes.label(xMarks2, yMarks2, Color.black, 2, Color.black)
+        axes.plot(Red, Color.red, "+")
+        axes.plot(Blue, Color.blue, "+")
+        axes.plot(Green, Color.green, "+")
+    }
+
+    init() {}
+
+    update() {}
+
+    draw(alpha) {}
+}
+
+var Game = Main.new()

Task/Goldbachs-comet/XPL0/goldbachs-comet.xpl0

@@ -0,0 +1,43 @@
+func IsPrime(N); \Return 'true' if N is prime
+int  N, I;
+[if N <= 2 then return N = 2;
+if (N&1) = 0 then \even >2\ return false;
+for I:= 3 to sqrt(N) do
+    [if rem(N/I) = 0 then return false;
+    I:= I+1;
+    ];
+return true;
+];
+
+int  PT(1_000_000);
+
+func G(E);      \Ways E can be expressed as sum of two primes
+int  E, C, I, J, T;
+[C:= 0;  I:= 0;
+loop    [J:= I;
+        if PT(J) + PT(I) > E then return C;
+        loop    [T:= PT(J) + PT(I);
+                if T = E then C:= C+1;
+                if T > E then quit;
+                J:= J+1;
+                ];
+        I:= I+1;
+        ];
+];
+
+int I, N;
+[I:= 0; \make prime table
+for N:= 2 to 1_000_000 do
+        if IsPrime(N) then
+                [PT(I):= N;  I:= I+1];
+I:= 4;  \show first 100 G numbers
+Format(4, 0);
+for N:= 1 to 100 do
+        [RlOut(0, float(G(I)));
+        if rem(N/10) = 0 then CrLf(0);
+        I:= I+2;
+        ];
+CrLf(0);
+Text(0, "G(1,000,000) = ");  IntOut(0, G(1_000_000));
+CrLf(0);
+]