Problem Statement

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.

161 is a Duffinian number.

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

Let's start with the solution:

  [ dup factors
    dup size 3 < iff
      [ 2drop false ] done
    0 swap witheach +
    gcd 1 = ]              is duffinian ( n --> b )

  [] 0
  [ dup duffinian if
      [ tuck join swap ]
    1+
    over size 50 = until ]
  drop
  [] swap
  witheach
    [ number$ nested join ]
  60 wrap$
  cr cr
  0 temp put
  [] 0
  [ dup duffinian iff
      [ 1 temp tally ]
    else
      [ 0 temp replace ]
    temp share 2 > if
      [ tuck 2 -
        join swap ]
    1+
    over size 15 = until ]
  drop
  [] swap
  witheach
    [ dup 1+ dup 1+
      join join
      nested join ]
  witheach [ echo cr ]