OFFSET
1,2
COMMENTS
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.
EXAMPLE
The sequence together with the corresponding compositions begins:
0: () 88: (2,1,4) 177: (2,1,4,1)
2: (2) 98: (1,4,2) 184: (2,1,1,4)
4: (3) 104: (1,2,4) 197: (1,4,2,1)
8: (4) 128: (8) 198: (1,4,1,2)
10: (2,2) 130: (6,2) 209: (1,2,4,1)
16: (5) 136: (4,4) 216: (1,2,1,4)
32: (6) 138: (4,2,2) 226: (1,1,4,2)
34: (4,2) 139: (4,2,1,1) 232: (1,1,2,4)
36: (3,3) 141: (4,1,2,1) 256: (9)
40: (2,4) 142: (4,1,1,2) 260: (6,3)
42: (2,2,2) 160: (2,6) 261: (6,2,1)
64: (7) 162: (2,4,2) 262: (6,1,2)
69: (4,2,1) 163: (2,4,1,1) 274: (4,3,2)
70: (4,1,2) 168: (2,2,4) 276: (4,2,3)
81: (2,4,1) 170: (2,2,2,2) 277: (4,2,2,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !CoprimeQ@@Union[stc[#]]&]
CROSSREFS
The complement is A333228.
Not ignoring repeated parts gives A335239.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Coprime partitions are counted by A327516.
Non-coprime partitions are counted by A335240.
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.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 28 2020
STATUS
approved