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2
Task/Meissel-Mertens-constant/00-META.yaml
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2
Task/Meissel-Mertens-constant/00-META.yaml
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---
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from: http://rosettacode.org/wiki/Meissel–Mertens_constant
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Task/Meissel-Mertens-constant/00-TASK.txt
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Task/Meissel-Mertens-constant/00-TASK.txt
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;Task:
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Calculate Meissel–Mertens constant up to a precision your language can handle.
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;Motivation:
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Analogous to Euler's constant, which is important in determining the sum of reciprocal natural numbers, Meissel-Mertens' constant is important in calculating the sum of reciprocal primes.
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;Example:
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We consider the finite sum of reciprocal natural numbers:
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''1 + 1/2 + 1/3 + 1/4 + 1/5 ... 1/n''
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this sum can be well approximated with:
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''log(n) + E''
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where ''E'' denotes Euler's constant: 0.57721...
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''log(n)'' denotes the natural logarithm of ''n''.
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Now consider the finite sum of reciprocal primes:
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''1/2 + 1/3 + 1/5 + 1/7 + 1/11 ... 1/p''
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this sum can be well approximated with:
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''log( log(p) ) + M''
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where ''M'' denotes Meissel-Mertens constant: 0.26149...
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;See:
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:* Details in the Wikipedia article: [https://en.wikipedia.org/wiki/Meissel%E2%80%93Mertens_constant Meissel–Mertens constant]
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<br /><br />
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@ -0,0 +1,22 @@
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F primes_up_to_limit(Int limit)
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[Int] r
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I limit >= 2
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r.append(2)
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V isprime = [1B] * ((limit - 1) I/ 2)
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V sieveend = Int(sqrt(limit))
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L(i) 0 .< isprime.len
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I isprime[i]
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Int p = i * 2 + 3
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r.append(p)
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I i <= sieveend
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L(j) ((p * p - 3) >> 1 .< isprime.len).step(p)
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isprime[j] = 0B
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R r
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V euler = 0.57721566490153286
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V m = 0.0
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L(x) primes_up_to_limit(10'000'000)
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m += log(1 - (1 / x)) + (1 / x)
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print(‘MM = #.16’.format(euler + m))
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@ -0,0 +1,45 @@
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BEGIN # compute an approximation to the Meissel-Mertens constant #
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# construct a sieve of odd primes #
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[ 0 : 10 000 000 ]BOOL primes;
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BEGIN
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FOR i TO UPB primes DO primes[ i ] := TRUE OD;
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INT ip := 1;
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FOR i WHILE i + ( ip +:= 2 ) <= UPB primes DO
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IF primes[ i ] THEN
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FOR s FROM i + ip BY ip TO UPB primes DO primes[ s ] := FALSE OD
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FI
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OD
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END;
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# sum the reciprocals of the primes #
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INT p count := 1;
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INT last p := 0;
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LONG REAL sum := long ln( 0.5 ) + 0.5;
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INT p := 1;
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INT p10 := 10;
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# Euler's constant from the wikipedia, truncated for LONG REAL #
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LONG REAL eulers constant = 0.5772156649015328606 # 0651209008240243104215933593992 #;
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FOR i TO UPB primes DO
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p +:= 2;
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IF primes[ i ] THEN
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LONG REAL rp = 1 / LENG p;
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sum +:= long ln( 1 - rp ) + rp;
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p count +:= 1;
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last p := p;
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IF p count = p10 THEN
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print( ( "after ", whole( p count, -8 ), " primes, the approximation is: "
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, fixed( sum + eulers constant, -14, 12 )
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, ", last prime considered: ", whole( last p, 0 )
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, newline
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)
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);
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p10 := IF p10 < 1 000 000 THEN p10 * 10 ELSE p10 + 1 000 000 FI
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FI
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FI
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OD;
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print( ( "after ", whole( p count, -8 ), " primes, the approximation is: "
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, fixed( sum + eulers constant, -14, 12 )
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, ", last prime considered: ", whole( last p, 0 )
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, newline
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)
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)
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END
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meisselMertens: function [depth][
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Euler: 0.57721566490153286
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m: (1//2) + ln 1-1//2
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loop range.step:2 3 depth 'x ->
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if prime? x ->
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m: m + (1//x) + ln 1-1//x
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return m + Euler
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]
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print meisselMertens 10000000
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Euler = 0.5772156649
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m = 0
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for x = 2 to 1e6 # more prime numbers do not add more precision
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if isPrime(x) then m += log(1-(1/x)) + (1/x)
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next x
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print "MM = "; Euler + m
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print Euler
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end
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function isPrime(v)
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if v < 2 then return False
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if v mod 2 = 0 then return v = 2
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if v mod 3 = 0 then return v = 3
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d = 5
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while d * d <= v
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if v mod d = 0 then return False else d += 2
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end while
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return True
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end function
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import 'dart:math';
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bool isPrime(var n) {
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if (n <= 1) return false;
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if (n == 2) return true;
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for (var i = 2; i <= sqrt(n); i++) if (n % i == 0) return false;
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return true;
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}
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void main() {
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const double euler = 0.57721566490153286;
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double m = 0.0;
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for (var x = 2; x <= 1e8; x++)
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if (isPrime(x)) m += log(1 - (1 / x)) + (1 / x);
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print('MM = ${euler + m}');
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}
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function IsPrime(N: int64): boolean;
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{Fast, optimised prime test}
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var I,Stop: int64;
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begin
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if (N = 2) or (N=3) then Result:=true
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else if (n <= 1) or ((n mod 2) = 0) or ((n mod 3) = 0) then Result:= false
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else
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begin
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I:=5;
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Stop:=Trunc(sqrt(N+0.0));
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Result:=False;
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while I<=Stop do
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begin
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if ((N mod I) = 0) or ((N mod (I + 2)) = 0) then exit;
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Inc(I,6);
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end;
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Result:=True;
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end;
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end;
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function GetNextPrime(var Start: integer): integer;
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{Get the next prime number after Start}
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{Start is passed by "reference," so the
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{original variable is incremented}
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begin
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repeat Inc(Start)
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until IsPrime(Start);
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Result:=Start;
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end;
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function MeisselMertens(Depth: integer; Prog: TProgress): extended;
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{Calculate MM value up a certain Depth}
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var I,P: integer;
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const Euler = 0.57721566490153286;
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begin
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Result:=0;
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P:=1;
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for I:=1 to Depth do
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begin
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P:=GetNextPrime(P);
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Result:=Result+Ln(1-(1/P)) + (1/P);
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if Assigned(Prog) and ((I mod 10000)=0) then
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HandleProgress(MulDiv(100,I,Depth));
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end;
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Result:=Result+Euler;
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end;
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procedure ShowMeisselMertens(Memo: TMemo; Prog: TProgress);
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var I,IT,Digits: integer;
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var M,Last: extended;
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begin
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Memo.Lines.Add('Primes Digits M');
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Memo.Lines.Add('-----------------------------------------');
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Last:=0;
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IT:=1;
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{Calculate MM to specified Power of 10}
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for I:=1 to 7 do
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begin
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IT:=IT*10;
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M:=MeisselMertens(IT,Prog);
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{Calculated Digits of accuracy}
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Digits:=Trunc(abs(Log(abs(M-Last))));
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Memo.Lines.Add(Format('10^%2d %7d %25.18f',[I,Digits,M]));
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Last:=M
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end;
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end;
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'#include "isprime.bas"
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Const As Double Euler = 0.57721566490153286
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Dim As Double m = 0
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For x As Ulongint = 2 To 1e8
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If isPrime(x) Then m += Log(1-(1/x)) + (1/x)
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Next x
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Print "MM ="; Euler + m
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Sleep
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33
Task/Meissel-Mertens-constant/Go/meissel-mertens-constant.go
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33
Task/Meissel-Mertens-constant/Go/meissel-mertens-constant.go
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package main
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import (
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"fmt"
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"math"
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"rcu"
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)
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func contains(a []int, f int) bool {
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for _, e := range a {
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if e == f {
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return true
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}
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}
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return false
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}
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func main() {
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const euler = 0.57721566490153286
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primes := rcu.Primes(1 << 31)
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pc := len(primes)
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sum := 0.0
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fmt.Println("Primes added M")
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fmt.Println("------------ --------------")
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for i, p := range primes {
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rp := 1.0 / float64(p)
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sum += math.Log(1.0-rp) + rp
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c := i + 1
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if (c%1e7) == 0 || c == pc {
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fmt.Printf("%11s %0.12f\n", rcu.Commatize(c), sum+euler)
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}
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}
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}
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Euler=: 0.57721566490153286
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MM=: {{ Euler + +/ (+ ^.@-.)@% p: i. y }}
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@ -0,0 +1,2 @@
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0j13 ": MM 1e8
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0.2614972128591
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mm=: (% 10x(^#)10#.inv]) 26149721284764278375542683860869585905156664826119920619206421392x
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@ -0,0 +1,2 @@
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0j65":mm
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0.26149721284764278375542683860869585905156664826119920619206421392
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@ -0,0 +1,41 @@
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import java.util.ArrayList;
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import java.util.BitSet;
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import java.util.List;
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/**
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* Calculates the Meissel-Mertens constant correct to 9 s.f. in approximately 15 seconds.
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*/
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public final class MeisselMertensConstant {
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public static void main(String[] aArgs) {
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List<Double> primeReciprocals = listPrimeReciprocals(1_000_000_000);
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final double euler = 0.577_215_664_901_532_861;
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double sum = 0.0;
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for ( double reciprocal : primeReciprocals ) {
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sum += reciprocal + Math.log(1.0 - reciprocal);
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}
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final double constant = euler + sum;
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System.out.println("The Meissel-Mertens constant = " + constant);
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}
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private static List<Double> listPrimeReciprocals(int aLimit) {
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BitSet sieve = new BitSet(aLimit + 1);
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sieve.set(2, aLimit + 1);
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final int squareRoot = (int) Math.sqrt(aLimit);
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for ( int i = 2; i <= squareRoot; i = sieve.nextSetBit(i + 1) ) {
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for ( int j = i * i; j <= aLimit; j += i ) {
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sieve.clear(j);
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}
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}
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List<Double> result = new ArrayList<Double>(sieve.cardinality());
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for ( int i = 2; i >= 0; i = sieve.nextSetBit(i + 1) ) {
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result.add(1.0 / i);
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}
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return result;
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}
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}
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@ -0,0 +1,9 @@
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using Base.MathConstants # sets constant γ = 0.5772156649015...
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using Primes
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""" Approximate the Meissel-Mertons constant. """
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function meissel_mertens(iterations = 100_000_000)
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return mapreduce(p ->(d = 1/p; log(1 - d) + d), +, primes(prime(iterations))) + γ
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end
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@show meissel_mertens(100_000_000) # meissel_mertens(100000000) = 0.2614972128591237
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@ -0,0 +1,4 @@
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PrimeNumbers = Select[Range[100000000], PrimeQ[#] &]; (*all primes in the first 100 000 000 numbers, this takes a toll on my computer's CPU and RAM*)
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MM = N[Total[Log[1 - 1/PrimeNumbers] + 1/PrimeNumbers] + EulerGamma, 10] (*Calculating it up to a precision of 10, this is correct up to 8 digits*)
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AnalyticMMto305 = N[EulerGamma + Sum[MoebiusMu[n]/n Log[Zeta[n]], {n, 2, 1000}], 1000] (*Precise up to 305 digits*)
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AnalyticMM = N[EulerGamma + Sum[MoebiusMu[n]/n Log[Zeta[n]], {n, 2, 10000}], 1001] (*Precise up to at least 1000 digits*)
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import std/[math, strformat, strutils]
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proc initPrimes(N: static int): seq[int] =
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## Initialize the list of primes.
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const M = 2 * N - 1
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var composite = newSeq[bool](N)
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composite[0] = true # 1 is not prime.
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# Conversions from index to value and value to index.
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template index(n: int): int = (n - 1) shr 1
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template value(idx: int): int = idx shl 1 + 1
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# Fill the sieve.
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var n = 3
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while n * n <= M:
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if not composite[n.index]:
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for k in countup(n * n, M, 2 * n):
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composite[k.index] = true
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inc n, 2
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# Build list of primes.
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result = @[2]
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for idx in 0..composite.high:
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if not composite[idx]:
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result.add idx.value
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const N = 2^30
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let primes = initPrimes(N)
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echo "Primes added M"
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echo "──────────── ──────────────"
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const γ = 0.57721566490153286 # Euler–Mascheroni constant.
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let primeCount = primes.len
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var sum = 0.0
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var count = 0
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for p in primes:
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let rp = 1 / p
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sum += ln(1 - rp) + rp
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inc count
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if count mod 10_000_000 == 0 or count == primeCount:
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echo &"{insertSep($count):>11} {sum+γ:.12}"
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@ -0,0 +1,8 @@
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{
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MM(t)=
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my(s=0);
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forprime(p = 2, t,
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s += log(1.-1./p)+1./p
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);
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Euler+s
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};
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@ -0,0 +1,16 @@
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{
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Meissel_Mertens(d)=
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default(realprecision, d);
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my(prec = default(realprecision), z = 0, y = 0, q);
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forprime(p = 2 , 7,
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z += log(1.-1./p)+1./p
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);
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for(k = 2, prec,
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q = 1;
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forprime(p = 2, 7,
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q *= 1.-p^-k
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);
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y += moebius(k)*log(zeta(k)*q)/k
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);
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Euler+z+y
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};
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@ -0,0 +1,6 @@
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use v5.36;
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use ntheory 'is_prime';
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my $s;
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is_prime $_ and $s += log(1 - 1/$_)+1/$_ for 2 .. 10**9;
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say my $result = $s + .57721566490153286;
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@ -0,0 +1,33 @@
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(phixonline)-->
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<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span> <span style="color: #000080;font-style:italic;">-- (but perhaps a bit too slow)</span>
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<span style="color: #008080;">constant</span> <span style="color: #000000;">mmc</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0.2614972128476427837554268386086958590516</span><span style="color: #0000FF;">,</span>
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<span style="color: #000000;">smmc</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0.2614972128476427837554268386086958590516"</span>
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<span style="color: #004080;">atom</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0.57721566490153286</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">dpa</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p10</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span>
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<span style="color: #004080;">integer</span> <span style="color: #000000;">pn</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">adp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">st</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"0"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">fmt</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"%.0f"</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;">"Primes added M\n"</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;">"------------ --------------\n"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">while</span> <span style="color: #000000;">adp</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">10</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">rp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #7060A8;">get_prime</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pn</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">t</span> <span style="color: #0000FF;">+=</span> <span style="color: #7060A8;">log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">rp</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">-</span><span style="color: #000000;">mmc</span><span style="color: #0000FF;">)<</span><span style="color: #000000;">dpa</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">ft</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fmt</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">trunc</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">p10</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">p10</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (as below)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">ft</span><span style="color: #0000FF;">=</span><span style="color: #000000;">st</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000080;font-style:italic;">--
|
||||
-- We have to artifically calculate a "truncated t", aka tt,
|
||||
-- to prevent say 0.2..299[>5..] being automatically rounded
|
||||
-- by printf() to 0.2..300, otherwise it just "looks wrong".
|
||||
--</span>
|
||||
<span style="color: #004080;">atom</span> <span style="color: #000000;">tt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trunc</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">1e12</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">1e12</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;">"%,11d %0.12f (accurate to %d d.p.)\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pn</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">tt</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">adp</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #000000;">adp</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
|
||||
<span style="color: #000000;">fmt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%%.%df"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">adp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">st</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">smmc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">2</span><span style="color: #0000FF;">+</span><span style="color: #000000;">adp</span><span style="color: #0000FF;">]</span>
|
||||
<span style="color: #000000;">dpa</span> <span style="color: #0000FF;">/=</span> <span style="color: #000000;">10</span>
|
||||
<span style="color: #000000;">p10</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">10</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #000000;">pn</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">while</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;">"(actual value 0.26149721284764278375542683860869)\n"</span><span style="color: #0000FF;">)</span>
|
||||
<!--
|
||||
|
|
@ -0,0 +1,54 @@
|
|||
-->
|
||||
<span style="color: #008080;">without</span> <span style="color: #008080;">js</span> <span style="color: #000080;font-style:italic;">-- no mpfr_zeta[_ui]() in mpfr.js, as yet, or mpfr_log() or mpfr_const_euler(), for that matter</span>
|
||||
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">moebius</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #004080;">sequence</span> <span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">prime_factors</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #004600;">true</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;">2</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">f</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;">f</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #7060A8;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">odd</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">))?-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">:+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">Meissel_Mertens</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_set_default_precision</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (d decimal places)</span>
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpfr_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">5</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">p</span> <span style="color: #008080;">in</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;">5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">7</span><span style="color: #0000FF;">}</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #000080;font-style:italic;">-- z += log(1-1/p)+1/p</span>
|
||||
<span style="color: #7060A8;">mpfr_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_div_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_si_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">d</span> <span style="color: #008080;">do</span> <span style="color: #000080;font-style:italic;">-- (see note)</span>
|
||||
<span style="color: #004080;">integer</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">moebius</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">m</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #7060A8;">mpfr_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</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;">p</span> <span style="color: #008080;">in</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;">5</span><span style="color: #0000FF;">,</span><span style="color: #000000;">7</span><span style="color: #0000FF;">}</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #000080;font-style:italic;">-- q *= 1-power(p,-k)</span>
|
||||
<span style="color: #7060A8;">mpfr_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_pow_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_si_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #000080;font-style:italic;">-- y += moebius(k)*log(zeta(k)*q)/k</span>
|
||||
<span style="color: #7060A8;">mpfr_zeta_ui</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_log</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_div_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rp</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<span style="color: #000080;font-style:italic;">-- res := EULER+z+y</span>
|
||||
<span style="color: #000000;">mpfr_const_euler</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpfr_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #004080;">mpfr</span> <span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">Meissel_Mertens</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1001</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #0000FF;">?</span><span style="color: #7060A8;">shorten</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mpfr_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1001</span><span style="color: #0000FF;">))</span>
|
||||
<!--
|
||||
|
|
@ -0,0 +1,30 @@
|
|||
Procedure isPrime(v.i)
|
||||
If v <= 1 : ProcedureReturn #False
|
||||
ElseIf v < 4 : ProcedureReturn #True
|
||||
ElseIf v % 2 = 0 : ProcedureReturn #False
|
||||
ElseIf v < 9 : ProcedureReturn #True
|
||||
ElseIf v % 3 = 0 : ProcedureReturn #False
|
||||
Else
|
||||
Protected r = Round(Sqr(v), #PB_Round_Down)
|
||||
Protected f = 5
|
||||
While f <= r
|
||||
If v % f = 0 Or v % (f + 2) = 0
|
||||
ProcedureReturn #False
|
||||
EndIf
|
||||
f + 6
|
||||
Wend
|
||||
EndIf
|
||||
ProcedureReturn #True
|
||||
EndProcedure
|
||||
|
||||
OpenConsole()
|
||||
Euler.d = 0.5772156649 ;0153286
|
||||
|
||||
For x.i = 2 To 1e8
|
||||
If isPrime(x)
|
||||
m.d = m + Log(1-(1/x)) + (1/x)
|
||||
EndIf
|
||||
Next x
|
||||
PrintN("MM = " + StrD(Euler + m))
|
||||
PrintN(#CRLF$ + "--- terminado, pulsa RETURN---"): Input()
|
||||
CloseConsole()
|
||||
|
|
@ -0,0 +1,19 @@
|
|||
#!/usr/bin/python
|
||||
|
||||
from math import log
|
||||
|
||||
def isPrime(n):
|
||||
for i in range(2, int(n**0.5) + 1):
|
||||
if n % i == 0:
|
||||
return False
|
||||
return True
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
Euler = 0.57721566490153286
|
||||
m = 0
|
||||
for x in range(2, 10_000_000):
|
||||
if isPrime(x):
|
||||
m += log(1-(1/x)) + (1/x)
|
||||
|
||||
print("MM =", Euler + m)
|
||||
|
|
@ -0,0 +1,5 @@
|
|||
# 20221011 Raku programming solution
|
||||
|
||||
my $s;
|
||||
.is-prime and $s += log(1-1/$_)+1/$_ for 2 .. 10**8;
|
||||
say $s + .57721566490153286
|
||||
|
|
@ -0,0 +1,17 @@
|
|||
function isPrime(n)
|
||||
if n < 2 then isPrime = 0 : goto [exit]
|
||||
if n = 2 then isPrime = 1 : goto [exit]
|
||||
if n mod 2 = 0 then isPrime = 0 : goto [exit]
|
||||
isPrime = 1
|
||||
for i = 3 to int(n^.5) step 2
|
||||
if n mod i = 0 then isPrime = 0 : goto [exit]
|
||||
next i
|
||||
[exit]
|
||||
end function
|
||||
|
||||
e = 0.5772156
|
||||
|
||||
for x = 2 to 100000 ' more prime numbers do not add more precision
|
||||
if isPrime(x) then m = m + log(1-(1/x)) + (1/x)
|
||||
next x
|
||||
print "MM = "; e + m
|
||||
|
|
@ -0,0 +1,25 @@
|
|||
FUNCTION isPrime (n)
|
||||
IF n = 2 THEN
|
||||
LET isPrime = 1
|
||||
ELSEIF n <= 1 OR REMAINDER(n, 2) = 0 THEN
|
||||
LET isPrime = 0
|
||||
ELSE
|
||||
LET isPrime = 1
|
||||
FOR i = 3 TO INT(SQR(n)) STEP 2
|
||||
IF REMAINDER(n, i) = 0 THEN
|
||||
LET isPrime = 0
|
||||
EXIT FUNCTION
|
||||
END IF
|
||||
NEXT i
|
||||
END IF
|
||||
END FUNCTION
|
||||
|
||||
LET e = .5772156649
|
||||
|
||||
FOR x = 2 to 1e6 !more prime numbers do not add more precision
|
||||
IF isPrime(x) = 1 THEN
|
||||
LET m = m + LOG(1-(1/x)) + (1/x)
|
||||
END IF
|
||||
NEXT x
|
||||
PRINT "MM ="; e + m
|
||||
END
|
||||
|
|
@ -0,0 +1,16 @@
|
|||
import "./math" for Int
|
||||
import "./fmt" for Fmt
|
||||
|
||||
var euler = 0.57721566490153286
|
||||
var primes = Int.primeSieve(2.pow(31))
|
||||
var pc = primes.count
|
||||
var sum = 0
|
||||
var c = 0
|
||||
System.print("Primes added M")
|
||||
System.print("------------ --------------")
|
||||
for (p in primes) {
|
||||
var rp = 1/p
|
||||
sum = (1-rp).log + rp + sum
|
||||
c = c + 1
|
||||
if ((c % 1e7) == 0 || c == pc) Fmt.print("$,11d $0.12f", c, sum + euler)
|
||||
}
|
||||
|
|
@ -0,0 +1,51 @@
|
|||
import "./gmp" for Mpf
|
||||
import "./math" for Int
|
||||
import "./fmt" for Fmt
|
||||
|
||||
var isSquareFree = Fn.new { |n|
|
||||
var i = 2
|
||||
while (i * i <= n) {
|
||||
if (n%(i*i) == 0) return false
|
||||
i = (i > 2) ? i + 2 : i + 1
|
||||
}
|
||||
return true
|
||||
}
|
||||
|
||||
var mu = Fn.new { |n|
|
||||
if (n < 1) Fiber.abort("Argument must be a positive integer")
|
||||
if (n == 1) return 1
|
||||
var sqFree = isSquareFree.call(n)
|
||||
var factors = Int.primeFactors(n)
|
||||
if (sqFree && factors.count % 2 == 0) return 1
|
||||
if (sqFree) return -1
|
||||
return 0
|
||||
}
|
||||
|
||||
var meisselMertens = Fn.new { |d|
|
||||
Mpf.defaultPrec = d
|
||||
var z = Mpf.zero
|
||||
var y = Mpf.zero
|
||||
var r = Mpf.new()
|
||||
var q = Mpf.new()
|
||||
var t = Mpf.new()
|
||||
var m = Mpf.new()
|
||||
for (p in [2, 3, 5, 7]) {
|
||||
r.setUi(p).inv
|
||||
t.uiSub(1, r).log.add(r)
|
||||
z.add(t, z)
|
||||
}
|
||||
for (k in 2..d) {
|
||||
q.setUi(1)
|
||||
for (p in [2, 3, 5, 7]) {
|
||||
r.setUi(p).inv
|
||||
t.uiSub(1, r.pow(k))
|
||||
q.mul(t)
|
||||
}
|
||||
m.setSi(mu.call(k))
|
||||
t.zetaUi(k).mul(q).log.mul(m).div(k)
|
||||
y.add(t, y)
|
||||
}
|
||||
return Mpf.euler.add(z).add(y)
|
||||
}
|
||||
|
||||
Fmt.print("$20a", meisselMertens.call(3300).toString(1001))
|
||||
|
|
@ -0,0 +1,24 @@
|
|||
func IsPrime(N); \Return 'true' if N is prime
|
||||
int N, D;
|
||||
[if N < 2 then return false;
|
||||
if (N&1) = 0 then return N = 2;
|
||||
if rem(N/3) = 0 then return N = 3;
|
||||
D:= 5;
|
||||
while D*D <= N do
|
||||
[if rem(N/D) = 0 then return false;
|
||||
D:= D+2;
|
||||
if rem(N/D) = 0 then return false;
|
||||
D:= D+4;
|
||||
];
|
||||
return true;
|
||||
];
|
||||
|
||||
def Euler = 0.57721566490153286;
|
||||
real M; int P;
|
||||
[M:= 0.;
|
||||
for P:= 2 to 100_000_000 do
|
||||
if IsPrime(P) then
|
||||
M:= M + Ln(1. - 1./float(P)) + 1./float(P);
|
||||
Format(1, 16);
|
||||
Text(0, "MM = "); RlOut(0, Euler + M); CrLf(0);
|
||||
]
|
||||
|
|
@ -0,0 +1,18 @@
|
|||
e = 0.5772156
|
||||
|
||||
for x = 2 to 1e6 // more prime numbers do not add more precision
|
||||
if isPrime(x) m = m + log(1-(1/x)) + (1/x)
|
||||
next x
|
||||
print "MM = ", e + m
|
||||
end
|
||||
|
||||
sub isPrime(v)
|
||||
if v < 2 return False
|
||||
if mod(v, 2) = 0 return v = 2
|
||||
if mod(v, 3) = 0 return v = 3
|
||||
d = 5
|
||||
while d * d <= v
|
||||
if mod(v, d) = 0 then return False else d = d + 2 : fi
|
||||
wend
|
||||
return True
|
||||
end sub
|
||||
Loading…
Add table
Add a link
Reference in a new issue