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

Author

Gus Wiseman, Mar 27 2020

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

Mathematica

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