A333227 Numbers k such that the k-th composition in standard order is pairwise coprime, where a singleton is not coprime unless it is (1).

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 75, 77, 78, 79, 80, 83, 89, 92, 95, 96, 97, 99, 101, 102, 103, 105

Offset

1,2

Comments

This is the definition used for CoprimeQ in Mathematica.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

Links

Table of n, a(n) for n=1..66.

Example

The sequence together with the corresponding compositions begins:
   1: (1)          27: (1,2,1,1)      55: (1,2,1,1,1)
   3: (1,1)        28: (1,1,3)        56: (1,1,4)
   5: (2,1)        29: (1,1,2,1)      57: (1,1,3,1)
   6: (1,2)        30: (1,1,1,2)      59: (1,1,2,1,1)
   7: (1,1,1)      31: (1,1,1,1,1)    60: (1,1,1,3)
   9: (3,1)        33: (5,1)          61: (1,1,1,2,1)
  11: (2,1,1)      35: (4,1,1)        62: (1,1,1,1,2)
  12: (1,3)        37: (3,2,1)        63: (1,1,1,1,1,1)
  13: (1,2,1)      38: (3,1,2)        65: (6,1)
  14: (1,1,2)      39: (3,1,1,1)      66: (5,2)
  15: (1,1,1,1)    41: (2,3,1)        67: (5,1,1)
  17: (4,1)        44: (2,1,3)        68: (4,3)
  18: (3,2)        47: (2,1,1,1,1)    71: (4,1,1,1)
  19: (3,1,1)      48: (1,5)          72: (3,4)
  20: (2,3)        49: (1,4,1)        75: (3,2,1,1)
  23: (2,1,1,1)    50: (1,3,2)        77: (3,1,2,1)
  24: (1,4)        51: (1,3,1,1)      78: (3,1,1,2)
  25: (1,3,1)      52: (1,2,3)        79: (3,1,1,1,1)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 120], CoprimeQ@@stc[#]&]

Crossrefs

A different ranking of the same compositions is A326675.

Ignoring repeated parts gives A333228.

Let q(k) be the k-th composition in standard order:

- The terms of q(k) are row k of A066099.

- The sum of q(k) is A070939(k).

- The product of q(k) is A124758(k).

- q(k) has A124767(k) runs and A333381(k) anti-runs.

- The GCD of q(k) is A326674(k).

- The Heinz number of q(k) is A333219(k).

- The LCM of q(k) is A333226(k).

Coprime or singleton sets are ranked by A087087.

Strict compositions are ranked by A233564.

Constant compositions are ranked by A272919.

Relatively prime compositions appear to be ranked by A291166.

Normal compositions are ranked by A333217.

Cf. A000120, A029931, A048793, A096111, A114994, A225620, A228351.

Sequence in context: A291166 A333228 A161373 * A285510 A174415 A356844

Adjacent sequences: A333224 A333225 A333226 * A333228 A333229 A333230

Keyword

nonn

Author

Gus Wiseman, Mar 27 2020

Status

approved