Duffinian numbers

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

• Numbers Aplenty - Duffinian numbers
• OEIS:A003624 - Duffinian numbers: composite numbers k relatively prime to sigma(k)

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

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)