Duffinian numbers: Difference between revisions

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=={{header|ALGOL 68}}==
=={{header|ALGOL 68}}==
Constructs a table of divisor counts without doing any divisions.
<lang algol68>BEGIN # find Duffinian numbers: non-primes relatively prime to their divisor count #
<lang algol68>BEGIN # find Duffinian numbers: non-primes relatively prime to their divisor count #
INT max number := 500 000; # largest number we will consider #
INT max number := 500 000; # largest number we will consider #
Line 62: Line 63:
OR gcd( n, nds ) /= 1
OR gcd( n, nds ) /= 1
THEN
THEN
# n is prime or is n not is relatively prime to its divisor sum #
# n is prime or is not is relatively prime to its divisor sum #
ds[ n ] := 0
ds[ n ] := 0
FI
FI

Revision as of 18:59, 25 February 2022

Duffinian 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.

A Duffinian number is a composite number k that is relatively prime to its sigma sum σ.

The sigma sum of k is the sum of the divisors of k.


E.G.

161 is a Duffinian number.

  • It is composite. (7 × 23)
  • The sigma sum 192 (1 + 7 + 23 + 161) is relatively prime to 161.


Duffinian numbers are very common.

It is not uncommon for two consecutive integers to be Duffinian (a Duffinian twin) (8, 9), (35, 36), (49, 50), etc.

Less common are Duffinian triplets; three consecutive Duffinian numbers. (63, 64, 65), (323, 324, 325), etc.

Much, much less common are Duffinian quadruplets and quintuplets. The first Duffinian quintuplet is (202605639573839041, 202605639573839042, 202605639573839043, 202605639573839044, 202605639573839045).

It is not possible to have six consecutive Duffinian numbers


Task
  • Find and show the first 50 Duffinian numbers.
  • Find and show at least the first 15 Duffinian triplets.


See also



ALGOL 68

Constructs a table of divisor counts without doing any divisions. <lang algol68>BEGIN # find Duffinian numbers: non-primes relatively prime to their divisor count # 

   INT max number            := 500 000;      # largest number we will consider   #
   # iterative Greatest Common Divisor routine, returns the gcd of m and n        #
   PROC gcd = ( INT m, n )INT:
        BEGIN
           INT a := ABS m, b := ABS n;
           WHILE b /= 0 DO
               INT new a = b;
               b        := a MOD b;
               a        := new a
           OD;
           a
        END # gcd # ;
   # construct a table of the divisor counts                                      #
   [ 1 : max number ]INT ds; FOR i TO UPB ds DO ds[ i ] := 1 OD;
   FOR i FROM 2 TO UPB ds
       DO FOR j FROM i BY i TO UPB ds DO ds[ j ] +:= i OD
   OD;
   # set the divisor counts of non-Duffinian numbers to 0                         #
   ds[ 1 ] := 0;  # 1 is not Duffinian                                            #
   FOR n FROM 2 TO UPB ds DO
       IF INT nds = ds[ n ];
          nds = n + 1
       OR gcd( n, nds ) /= 1
       THEN
           # n is prime or is not is relatively prime to its divisor sum          #
           ds[ n ] := 0
       FI
   OD;
   # show the first 50 Duffinian numbers                                          #
   print( ( "The first 50 Duffinian numbers:", newline ) );
   INT dcount := 0;
   FOR n WHILE dcount < 50 DO
       IF ds[ n ] /= 0
       THEN # found a Duffinian number                                            #
           print( ( " ", whole( n, -3) ) );
           IF ( dcount +:= 1 ) MOD 25 = 0 THEN print( ( newline ) ) FI
       FI
   OD;
   print( ( newline ) );
   # show the duffinian triplets below UPB ds                                     #
   print( ( "The Duffinian triplets up to ", whole( UPB ds, 0 ), ":", newline ) );
   dcount := 0;
   FOR n FROM 3 TO UPB ds DO
       IF ds[ n - 2 ] /= 0 AND ds[ n - 1 ] /= 0 AND ds[ n ] /= 0
       THEN # found a Duffinian triplet                                           #
           print( ( " (", whole( n - 2, -7 ), " ", whole( n - 1, -7 ), " ", whole( n, -7 ), ")" ) );
           IF ( dcount +:= 1 ) MOD 4 = 0 THEN print( ( newline ) ) FI
       FI
   OD

END</lang>

Output:
The first 50 Duffinian numbers:
   4   8   9  16  21  25  27  32  35  36  39  49  50  55  57  63  64  65  75  77  81  85  93  98 100
 111 115 119 121 125 128 129 133 143 144 155 161 169 171 175 183 185 187 189 201 203 205 209 215 217

The Duffinian triplets up to 500000:
 (     63      64      65) (    323     324     325) (    511     512     513) (    721     722     723)
 (    899     900     901) (   1443    1444    1445) (   2303    2304    2305) (   2449    2450    2451)
 (   3599    3600    3601) (   3871    3872    3873) (   5183    5184    5185) (   5617    5618    5619)
 (   6049    6050    6051) (   6399    6400    6401) (   8449    8450    8451) (  10081   10082   10083)
 (  10403   10404   10405) (  11663   11664   11665) (  12481   12482   12483) (  13447   13448   13449)
 (  13777   13778   13779) (  15841   15842   15843) (  17423   17424   17425) (  19043   19044   19045)
 (  26911   26912   26913) (  30275   30276   30277) (  36863   36864   36865) (  42631   42632   42633)
 (  46655   46656   46657) (  47523   47524   47525) (  53137   53138   53139) (  58563   58564   58565)
 (  72961   72962   72963) (  76175   76176   76177) (  79523   79524   79525) (  84099   84100   84101)
 (  86527   86528   86529) (  94177   94178   94179) ( 108899  108900  108901) ( 121103  121104  121105)
 ( 125315  125316  125317) ( 128017  128018  128019) ( 129599  129600  129601) ( 137287  137288  137289)
 ( 144399  144400  144401) ( 144721  144722  144723) ( 154567  154568  154569) ( 158403  158404  158405)
 ( 166463  166464  166465) ( 167041  167042  167043) ( 175231  175232  175233) ( 177607  177608  177609)
 ( 181475  181476  181477) ( 186623  186624  186625) ( 188497  188498  188499) ( 197191  197192  197193)
 ( 199711  199712  199713) ( 202499  202500  202501) ( 211249  211250  211251) ( 230399  230400  230401)
 ( 231199  231200  231201) ( 232561  232562  232563) ( 236195  236196  236197) ( 242063  242064  242065)
 ( 243601  243602  243603) ( 248003  248004  248005) ( 260099  260100  260101) ( 260641  260642  260643)
 ( 272483  272484  272485) ( 274575  274576  274577) ( 285155  285156  285157) ( 291599  291600  291601)
 ( 293763  293764  293765) ( 300303  300304  300305) ( 301087  301088  301089) ( 318095  318096  318097)
 ( 344449  344450  344451) ( 354481  354482  354483) ( 359551  359552  359553) ( 359999  360000  360001)
 ( 367235  367236  367237) ( 374543  374544  374545) ( 403201  403202  403203) ( 406801  406802  406803)
 ( 417697  417698  417699) ( 419903  419904  419905) ( 423199  423200  423201) ( 435599  435600  435601)
 ( 468511  468512  468513) ( 470449  470450  470451) ( 488071  488072  488073)

Factor

Works with: Factor version 0.99 2021-06-02

<lang factor>USING: combinators.short-circuit.smart grouping io kernel lists lists.lazy math math.primes math.primes.factors math.statistics prettyprint sequences sequences.deep ;

duffinian? ( n -- ? )
   { [ prime? not ] [ dup divisors sum simple-gcd 1 = ] } && ;
duffinians ( -- list ) 3 lfrom [ duffinian? ] lfilter ;
triples ( -- list )
   duffinians dup cdr dup cdr lzip lzip [ flatten ] lmap-lazy
   [ differences { 1 1 } = ] lfilter ;

"First 50 Duffinian numbers:" print 50 duffinians ltake list>array 10 group simple-table. nl

"First 15 Duffinian triplets:" print 15 triples ltake list>array simple-table.</lang>

Output:
First 50 Duffinian numbers:
4   8   9   16  21  25  27  32  35  36
39  49  50  55  57  63  64  65  75  77
81  85  93  98  100 111 115 119 121 125
128 129 133 143 144 155 161 169 171 175
183 185 187 189 201 203 205 209 215 217

First 15 Duffinian triplets:
63   64   65
323  324  325
511  512  513
721  722  723
899  900  901
1443 1444 1445
2303 2304 2305
2449 2450 2451
3599 3600 3601
3871 3872 3873
5183 5184 5185
5617 5618 5619
6049 6050 6051
6399 6400 6401
8449 8450 8451

Julia

<lang julia>using Primes

function σ(n) 

   f = [one(n)]
   for (p,e) in factor(n)
       f = reduce(vcat, [f*p^j for j in 1:e], init=f)
   end
   return sum(f)

end

isDuffinian(n) = !isprime(n) && gcd(n, σ(n)) == 1

function testDuffinians() 

   println("First 50 Duffinian numbers:")
   foreach(p -> print(rpad(p[2], 4), p[1] % 25 == 0 ? "\n" : ""),
       enumerate(filter(isDuffinian, 2:217)))
   n, found = 2, 0
   println("\nFifteen Duffinian triplets:")
   while found < 15
       if isDuffinian(n) && isDuffinian(n + 1) && isDuffinian(n + 2)
           println(lpad(n, 6), lpad(n +1, 6), lpad(n + 2, 6))
           found += 1
       end
       n += 1
   end

end

testDuffinians()

</lang>

Output:
First 50 Duffinian numbers:
4   8   9   16  21  25  27  32  35  36  39  49  50  55  57  63  64  65  75  77  81  85  93  98  100 
111 115 119 121 125 128 129 133 143 144 155 161 169 171 175 183 185 187 189 201 203 205 209 215 217

Fifteen Duffinian triplets:
    63    64    65
   323   324   325
   511   512   513
   721   722   723
   899   900   901
  1443  1444  1445
  2303  2304  2305
  2449  2450  2451
  3599  3600  3601
  3871  3872  3873
  5183  5184  5185
  5617  5618  5619
  6049  6050  6051
  6399  6400  6401
  8449  8450  8451

Phix

with javascript_semantics
sequence duffinian = {false}
integer n = 2, count = 0, triplet = 0, triple_count = 0
while triple_count<50 do
    bool bDuff = not is_prime(n) and gcd(n,sum(factors(n,1)))=1
    duffinian &= bDuff
    if bDuff then
        count += 1
        if count=50 then
            sequence s50 = apply(true,sprintf,{{"%3d"},find_all(true,duffinian)})
            printf(1,"First 50 Duffinian numbers:\n%s\n",join_by(s50,1,25," "))
        end if
        triplet += 1
        triple_count += (triplet>=3)
    else
        triplet = 0
    end if
    n += 1
end while
sequence s = apply(true,sq_add,{match_all({true,true,true},duffinian),{{0,1,2}}}),
         p = apply(true,pad_tail,{apply(true,sprintf,{{"[%d,%d,%d]"},s}),24})
printf(1,"First 50 Duffinian triplets:\n%s\n",{join_by(p,1,4," ")})
Output:
First 50 Duffinian numbers:
  4   8   9  16  21  25  27  32  35  36  39  49  50  55  57  63  64  65  75  77  81  85  93  98 100
111 115 119 121 125 128 129 133 143 144 155 161 169 171 175 183 185 187 189 201 203 205 209 215 217

First 50 Duffinian triplets:
[63,64,65]               [323,324,325]            [511,512,513]            [721,722,723]
[899,900,901]            [1443,1444,1445]         [2303,2304,2305]         [2449,2450,2451]
[3599,3600,3601]         [3871,3872,3873]         [5183,5184,5185]         [5617,5618,5619]
[6049,6050,6051]         [6399,6400,6401]         [8449,8450,8451]         [10081,10082,10083]
[10403,10404,10405]      [11663,11664,11665]      [12481,12482,12483]      [13447,13448,13449]
[13777,13778,13779]      [15841,15842,15843]      [17423,17424,17425]      [19043,19044,19045]
[26911,26912,26913]      [30275,30276,30277]      [36863,36864,36865]      [42631,42632,42633]
[46655,46656,46657]      [47523,47524,47525]      [53137,53138,53139]      [58563,58564,58565]
[72961,72962,72963]      [76175,76176,76177]      [79523,79524,79525]      [84099,84100,84101]
[86527,86528,86529]      [94177,94178,94179]      [108899,108900,108901]   [121103,121104,121105]
[125315,125316,125317]   [128017,128018,128019]   [129599,129600,129601]   [137287,137288,137289]
[144399,144400,144401]   [144721,144722,144723]   [154567,154568,154569]   [158403,158404,158405]
[166463,166464,166465]   [167041,167042,167043]

Raku

<lang perl6>use Prime::Factor;

my @duffinians = lazy (3..*).hyper.grep: { !.is-prime && $_ gcd .&divisors.sum == 1 };

put "First 50 Duffinian numbers:\n" ~ @duffinians[^50].batch(10)».fmt("%3d").join: "\n";

put "\nFirst 40 Duffinian triplets:\n" ~ 

   ((^∞).grep: -> $n { (@duffinians[$n] + 1 == @duffinians[$n + 1]) && (@duffinians[$n] + 2 == @duffinians[$n + 2]) })[^40]\
   .map( { "({@duffinians[$_ .. $_+2].join: ', '})" } ).batch(4)».fmt("%-24s").join: "\n";</lang>
Output:
First 50 Duffinian numbers:
  4   8   9  16  21  25  27  32  35  36
 39  49  50  55  57  63  64  65  75  77
 81  85  93  98 100 111 115 119 121 125
128 129 133 143 144 155 161 169 171 175
183 185 187 189 201 203 205 209 215 217

First 40 Duffinian triplets:
(63, 64, 65)             (323, 324, 325)          (511, 512, 513)          (721, 722, 723)         
(899, 900, 901)          (1443, 1444, 1445)       (2303, 2304, 2305)       (2449, 2450, 2451)      
(3599, 3600, 3601)       (3871, 3872, 3873)       (5183, 5184, 5185)       (5617, 5618, 5619)      
(6049, 6050, 6051)       (6399, 6400, 6401)       (8449, 8450, 8451)       (10081, 10082, 10083)   
(10403, 10404, 10405)    (11663, 11664, 11665)    (12481, 12482, 12483)    (13447, 13448, 13449)   
(13777, 13778, 13779)    (15841, 15842, 15843)    (17423, 17424, 17425)    (19043, 19044, 19045)   
(26911, 26912, 26913)    (30275, 30276, 30277)    (36863, 36864, 36865)    (42631, 42632, 42633)   
(46655, 46656, 46657)    (47523, 47524, 47525)    (53137, 53138, 53139)    (58563, 58564, 58565)   
(72961, 72962, 72963)    (76175, 76176, 76177)    (79523, 79524, 79525)    (84099, 84100, 84101)   
(86527, 86528, 86529)    (94177, 94178, 94179)    (108899, 108900, 108901) (121103, 121104, 121105)

Wren

Library: Wren-math
Library: Wren-seq
Library: Wren-fmt

<lang ecmascript>import "./math" for Int, Nums import "./seq" for Lst import "./fmt" for Fmt

var limit = 200000 // say var d = Int.primeSieve(limit-1, false) d[1] = false for (i in 2...limit) { 

   if (!d[i]) continue
   if (i % 2 == 0 && !Int.isSquare(i) && !Int.isSquare(i/2)) {
       d[i] = false
       continue
   }
   var sigmaSum = Nums.sum(Int.divisors(i))
   if (Int.gcd(sigmaSum, i) != 1) d[i] = false

}

var duff = (1...d.count).where { |i| d[i] }.toList System.print("First 50 Duffinian numbers:") for (chunk in Lst.chunks(duff[0..49], 10)) Fmt.print("$3d", chunk)

var triplets = [] for (i in 2...limit) { 

   if (d[i] && d[i-1] && d[i-2]) triplets.add([i-2, i-1, i])

} System.print("\nFirst 50 Duffinian triplets:") for (chunk in Lst.chunks(triplets[0..49], 4)) Fmt.print("$-25s", chunk)</lang>

Output:
First 50 Duffinian numbers:
  4   8   9  16  21  25  27  32  35  36
 39  49  50  55  57  63  64  65  75  77
 81  85  93  98 100 111 115 119 121 125
128 129 133 143 144 155 161 169 171 175
183 185 187 189 201 203 205 209 215 217

First 50 Duffinian triplets:
[63, 64, 65]              [323, 324, 325]           [511, 512, 513]           [721, 722, 723]          
[899, 900, 901]           [1443, 1444, 1445]        [2303, 2304, 2305]        [2449, 2450, 2451]       
[3599, 3600, 3601]        [3871, 3872, 3873]        [5183, 5184, 5185]        [5617, 5618, 5619]       
[6049, 6050, 6051]        [6399, 6400, 6401]        [8449, 8450, 8451]        [10081, 10082, 10083]    
[10403, 10404, 10405]     [11663, 11664, 11665]     [12481, 12482, 12483]     [13447, 13448, 13449]    
[13777, 13778, 13779]     [15841, 15842, 15843]     [17423, 17424, 17425]     [19043, 19044, 19045]    
[26911, 26912, 26913]     [30275, 30276, 30277]     [36863, 36864, 36865]     [42631, 42632, 42633]    
[46655, 46656, 46657]     [47523, 47524, 47525]     [53137, 53138, 53139]     [58563, 58564, 58565]    
[72961, 72962, 72963]     [76175, 76176, 76177]     [79523, 79524, 79525]     [84099, 84100, 84101]    
[86527, 86528, 86529]     [94177, 94178, 94179]     [108899, 108900, 108901]  [121103, 121104, 121105] 
[125315, 125316, 125317]  [128017, 128018, 128019]  [129599, 129600, 129601]  [137287, 137288, 137289] 
[144399, 144400, 144401]  [144721, 144722, 144723]  [154567, 154568, 154569]  [158403, 158404, 158405] 
[166463, 166464, 166465]  [167041, 167042, 167043]

XPL0

<lang XPL0>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; ];

func SumDiv(Num); \Return sum of proper divisors of Num int Num, Div, Sum, Quot; [Div:= 2; Sum:= 0; loop [Quot:= Num/Div; 

       if Div > Quot then quit;
       if rem(0) = 0 then
           [Sum:= Sum + Div;
           if Div # Quot then Sum:= Sum + Quot;
           ];
       Div:= Div+1;
       ];

return Sum+1; ];

func GCD(A, B); \Return greatest common divisor of A and B int A, B; [while A#B do 

 if A>B then A:= A-B
        else B:= B-A;

return A; ];

func Duff(N); \Return 'true' if N is a Duffinian number int N; [if IsPrime(N) then return false; return GCD(SumDiv(N), N) = 1; ];

int C, N; [Format(4, 0); C:= 0; N:= 4; loop [if Duff(N) then 

           [RlOut(0, float(N));
           C:= C+1;
           if C >= 50 then quit;
           if rem(C/20) = 0 then CrLf(0);
           ];
       N:= N+1;
       ];

CrLf(0); CrLf(0); Format(5, 0); C:= 0; N:= 4; loop [if Duff(N) & Duff(N+1) & Duff(N+2) then 

           [RlOut(0, float(N));  RlOut(0, float(N+1));  RlOut(0, float(N+2));
           CrLf(0);
           C:= C+1;
           if C >= 15 then quit;
           ];
       N:= N+1;
       ];

]</lang>

Output:
   4   8   9  16  21  25  27  32  35  36  39  49  50  55  57  63  64  65  75  77
  81  85  93  98 100 111 115 119 121 125 128 129 133 143 144 155 161 169 171 175
 183 185 187 189 201 203 205 209 215 217

   63   64   65
  323  324  325
  511  512  513
  721  722  723
  899  900  901
 1443 1444 1445
 2303 2304 2305
 2449 2450 2451
 3599 3600 3601
 3871 3872 3873
 5183 5184 5185
 5617 5618 5619
 6049 6050 6051
 6399 6400 6401
 8449 8450 8451
Retrieved from "https://rosettacode.org/wiki/Duffinian_numbers?oldid=319927"