Giuga numbers

Giuga numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Definition

A Giuga number is a composite number n which is such that each of its distinct prime factors f divide (n/f - 1) exactly.

All known Giuga numbers are even though it is not known for certain that there are no odd examples.

Example

30 is a Giuga number because its distinct prime factors are 2, 3 and 5 and:

• 30/2 - 1 = 14 is divisible by 2
• 30/3 - 1 = 9 is divisible by 3
• 30/5 - 1 = 5 is divisible by 5

Task

Determine and show here the first four Giuga numbers.

Stretch

Determine the fifth Giuga number and any more you have the patience for.

References

• Wikipedia: Giuga number
• OEIS:A007850 - Giuga numbers

FreeBASIC

<lang freebasic>Function isGiuga(m As Uinteger) As Boolean

   Dim As Uinteger n = m
   Dim As Uinteger f = 2, l = Sqr(n)
   Do
       If n Mod f = 0 Then
           If ((m / f) - 1) Mod f <> 0 Then Return False
           n /= f
           If f > n Then Return True
       Else    
           f += 1
           If f > l Then Return False
       End If
   Loop

End Function

Dim As Uinteger n = 3, c = 0, limit = 4 Print "The first "; limit; " Giuga numbers are: "; Do

   If isGiuga(n) Then c += 1: Print n; "  ";
   n += 1

Loop Until c = limit</lang>

Output:

The first 4 Giuga numbers are: 30  858  1722  66198

Go

Translation of: Wren

Library: Go-rcu

I thought I'd see how long it would take to 'brute force' the fifth Giuga number and the answer (without using parallelization, Core i7) is around 2 hours 40 minutes. <lang go>package main

import (

   "fmt"
   "rcu"

)

func main() {

   const limit = 5
   var giuga []int
   for n := 4; len(giuga) < limit; n += 2 {
       factors := rcu.PrimeFactors(n)
       if len(factors) > 2 {
           isSquareFree := true
           for i := 1; i < len(factors); i++ {
               if factors[i] == factors[i-1] {
                   isSquareFree = false
                   break
               }
           }
           if isSquareFree {
               isGiuga := true
               for _, f := range factors {
                   if (n/f-1)%f != 0 {
                       isGiuga = false
                       break
                   }
               }
               if isGiuga {
                   giuga = append(giuga, n)
               }
           }
       }
   }
   fmt.Println("The first", limit, "Giuga numbers are:")
   fmt.Println(giuga)

}</lang>

Output:

The first 5 Giuga numbers are:
[30 858 1722 66198 2214408306]

J

We can brute force this task building a test for giuga numbers and checking the first hundred thousand integers (which takes a small fraction of a second):

<lang J>giguaP=: {{ (1<y)*(-.1 p:y)**/(=<.) y ((_1+%)%]) q: y }}"0</lang>

<lang J> 1+I.giguaP 1+i.1e5 30 858 1722 66198</lang>

These numbers have some interesting properties but there's an issue with guaranteeing correctness of more sophisticated approaches.

Julia

<lang ruby>using Primes

isGiuga(n) = all(f -> f != n && rem(n ÷ f - 1, f) == 0, factor(Vector, n))

function getGiuga(N)

   gcount = 0
   for i in 4:typemax(Int)
       if isGiuga(i)
           println(i)
           (gcount += 1) >= N && break
       end
   end

end

getGiuga(4)

</lang>

Output:

30      
858 
1722
66198

Pascal

Free Pascal

changing main routine at the end of Factors_of_an_integer#using_Prime_decomposition
Mostly, about 70%, the highest primefactor remains in pfRemain.
<lang pascal> function ChkGuiga(n:Uint64;pPrimeDecomp :tpPrimeFac):boolean;inline; var

 pFr : Uint64;
 idx: NativeInt;
 p : Uint32;

begin

with pPrimeDecomp^ do
Begin
 pfR := pfRemain;
 idx := pfMaxIdx;//always>0 for pfDivcnt>4
 //check squarefree. i primefactors -> count of diviors = 2^i 
 p := 1 shl idx;
 if pfR <> 1 then
 begin
   if (p+p <> pfdivcnt) then
     EXIT(false);
   if sqr(pfR)>=n then
     EXIT(false);
   //squarefree
   result := (((n DIV pfR)-1)MOD pfR) = 0;
   if result then
   begin
     idx -= 1;
     repeat
       p := SmallPrimes[pfpotPrimIdx[idx]];
       result := (((n DIV p)-1)MOD p) = 0;
       if Not(result) then
         EXIT(result);
       dec(idx);
     until idx <0;
   end;
 end
 else
 begin
   if (p <> pfdivcnt) then
     EXIT(false);
   result := true;
   repeat
     dec(idx);
     p := SmallPrimes[pfpotPrimIdx[idx]];
     result := (((n DIV p)-1)MOD p) = 0;
     if not(result) then
       EXIT(false);
   until idx<=0;
 end;
end;

end;

const

 LMT = 24423128562;//24423128562;//2214408306;//

var

 pPrimeDecomp :tpPrimeFac;
 T0:Int64;
 n : NativeUInt;
 cnt:Uint32;

Begin

 InitSmallPrimes;

 T0 := GetTickCount64;
 Init_Sieve(1);
 n := 0;
 cnt := 0;
 repeat
   //only even numbers
   inc(n,2);
   GetNextPrimeDecomp;
   pPrimeDecomp:= GetNextPrimeDecomp;
   //if not prime/semiprime
   if pPrimeDecomp^.pfDivCnt >4 then
     if ChkGuiga(n,pPrimeDecomp) then
     begin
       inc(cnt);
       writeln(cnt:3,'..',OutPots(pPrimeDecomp,n),'  ',(GettickCount64-T0)/1000:6:3,' s');
     end;
 until n >= LMT;
 T0 := GetTickCount64-T0;
 writeln('Found ',cnt);
 writeln('Tested til ',n,' runtime ',T0/1000:0:3,' s');
 writeln;
 writeln(OutPots(pPrimeDecomp,n));

end.</lang>

@home AMD 5600G fpc3.2.2 -O4 -Xs:

  1..30 :  8 : 2*3*5_chk_30<_SoD_72<   0.001 s
  2..858 : 16 : 2*3*11*13_chk_858<_SoD_2016<   0.001 s
  3..1722 : 16 : 2*3*7*41_chk_1722<_SoD_4032<   0.001 s
  4..66198 : 32 : 2*3*11*17*59_chk_66198<_SoD_155520<   0.002 s
  5..2214408306 : 64 : 2*3*11*23*31*47057_chk_2214408306<_SoD_5204238336<  41.656 s
  6..24423128562 : 64 : 2*3*7*43*3041*4447_chk_24423128562<_SoD_57154166784<  533.047 s
Found 6
Tested til 24423128562 runtime 533.047 s

Perl

<lang perl>#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/Giuga_numbers use warnings; use ntheory qw( factor forcomposites ); use List::Util qw( all );

forcomposites

 {
 my $n = $_;
 all { ($n / $_ - 1) % $_ == 0 } factor $n and print "$n\n";
 } 4, 67000;</lang>

Output:

30
858
1722
66198

Phix

with javascript_semantics
constant limit = 4
sequence giuga = {}
integer n = 4
while length(giuga)<limit do
    sequence pf = prime_factors(n)
    for f in pf do
        if remainder(n/f-1,f) then pf={} exit end if
    end for
    if length(pf) then giuga &= n end if
    n += 2
end while
printf(1,"The first %d Giuga numbers are: %v\n",{limit,giuga})

Output:

The first 4 Giuga numbers are: {30,858,1722,66198}

Python

Translation of: FreeBASIC

<lang python>#!/usr/bin/python

from math import sqrt

def isGiuga(m):

   n = m
   f = 2
   l = sqrt(n)
   while True:
       if n % f == 0:
           if ((m / f) - 1) % f != 0:
               return False
           n /= f
           if f > n:
               return True
       else:
           f += 1
           if f > l:
               return False

if __name__ == '__main__':

   n = 3
   c = 0
   print("The first 4 Giuga numbers are: ")
   while c < 4:
       if isGiuga(n):
           c += 1
           print(n)
       n += 1</lang>

Raku

<lang perl6>my @primes = (3..60).grep: &is-prime;

print 'First four Giuga numbers: ';

put sort flat (2..4).map: -> $c {

   @primes.combinations($c).map: {
       my $n = [×] 2,|$_;
       $n if all .map: { ($n / $_ - 1) %% $_ };
   }

}</lang>

Output:

First 4 Giuga numbers: 30 858 1722 66198

Wren

Library: Wren-math

Library: Wren-seq

Simple brute force but assumes all Giuga numbers will be even, must be square-free and can't be semi-prime.

Takes only about 0.1 seconds to find the first four Giuga numbers but finding the fifth would take many hours using this approach, so I haven't bothered. Hopefully, someone can come up with a better method. <lang ecmascript>import "./math" for Int import "./seq" for Lst

var limit = 4 var giuga = [] var n = 4 while (giuga.count < limit) {

   var factors = Int.primeFactors(n)
   var c = factors.count
   if (c > 2) {
       var factors2 = Lst.prune(factors)
       if (c == factors2.count && factors2.all { |f| (n/f - 1) % f == 0 }) {
           giuga.add(n)
       }
   }
   n = n + 2

} System.print("The first %(limit) Giuga numbers are:") System.print(giuga)</lang>

Output:

The first 4 Giuga numbers are:
[30, 858, 1722, 66198]

XPL0

<lang XPL0>func Giuga(N0); \Return 'true' if Giuga number int N0; int N, F, Q1, Q2, L; [N:= N0; F:= 2; L:= sqrt(N); loop [Q1:= N/F;

       if rem(0) = 0 then      \found a prime factor
               [Q2:= N0/F;
               if rem((Q2-1)/F) # 0 then return false;
               N:= Q1;
               if F>N then quit;
               ]
       else    [F:= F+1;
               if F>L then return false;
               ];
       ];

return true; ];

int N, C; [N:= 3; C:= 0; loop [if Giuga(N) then

               [IntOut(0, N);  ChOut(0, ^ );
               C:= C+1;
               if C >= 4 then quit;
               ];
       N:= N+1;
       ];

]</lang>

Output:

30 858 1722 66198