A333228 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Author

Gus Wiseman, May 28 2020

Keywords

• nonn

Comments

First differs from A291166 in lacking 69, which corresponds to the composition (4,2,1).

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The sequence together with the corresponding compositions begins:
   1: (1)          21: (2,2,1)        39: (3,1,1,1)
   3: (1,1)        22: (2,1,2)        41: (2,3,1)
   5: (2,1)        23: (2,1,1,1)      43: (2,2,1,1)
   6: (1,2)        24: (1,4)          44: (2,1,3)
   7: (1,1,1)      25: (1,3,1)        45: (2,1,2,1)
   9: (3,1)        26: (1,2,2)        46: (2,1,1,2)
  11: (2,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
  12: (1,3)        28: (1,1,3)        48: (1,5)
  13: (1,2,1)      29: (1,1,2,1)      49: (1,4,1)
  14: (1,1,2)      30: (1,1,1,2)      50: (1,3,2)
  15: (1,1,1,1)    31: (1,1,1,1,1)    51: (1,3,1,1)
  17: (4,1)        33: (5,1)          52: (1,2,3)
  18: (3,2)        35: (4,1,1)        53: (1,2,2,1)
  19: (3,1,1)      37: (3,2,1)        54: (1,2,1,2)
  20: (2,3)        38: (3,1,2)        55: (1,2,1,1,1)
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

Pairwise coprime or singleton partitions are A051424.

Coprime or singleton sets are ranked by A087087.

The version for relatively prime instead of coprime appears to be A291166.

Numbers whose binary indices are pairwise coprime are A326675.

Coprime partitions are counted by A327516.

Not ignoring repeated parts gives A333227.

The complement is A335238.

All of the following pertain to compositions in standard order (A066099):

- Length is A000120.

- Sum is A070939.

- Product is A124758.

- Reverse is A228351

- GCD is A326674.

- Heinz number is A333219.

- LCM is A333226.

- Number of distinct parts is A334028.

Cf. A007360, A048793, A101268, A233564, A272919, A291166, A302569, A335235, A335236, A335237, A335239.

Programs

Mathematica

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