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 greater than 2, 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 quintuplets - five consecutive Duffinian numbers; the first of which is (202605639573839041, 202605639573839042, 202605639573839043, 202605639573839044, 202605639573839045).

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)

Raku

<lang perl6>use Prime::Factor;

sub is-Duffinian ($_) { !.is-prime && $_ gcd .&divisors.sum == 1 }

put "First 50 Duffinian numbers:\n" ~ (3..*).hyper.grep(&is-Duffinian)[^50].batch(10)».fmt("%3d").join: "\n";

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

   ((3..*).hyper.grep: -> $n { all (^3).map: { ($_ + $n).&is-Duffinian } })[^40]\
   .map( { "({($_..$_+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)