A335239 Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).

0, 2, 4, 8, 10, 16, 21, 22, 26, 32, 34, 36, 40, 42, 43, 45, 46, 53, 54, 58, 64, 69, 70, 73, 74, 76, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 98, 100, 104, 106, 107, 109, 110, 117, 118, 122, 128, 130, 136, 138, 139, 141, 142, 146, 147, 148, 149, 150, 153

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.

Links

Table of n, a(n) for n=1..60.

Example

The sequence together with the corresponding compositions begins:
    0: ()            45: (2,1,2,1)     86: (2,2,1,2)
    2: (2)           46: (2,1,1,2)     87: (2,2,1,1,1)
    4: (3)           53: (1,2,2,1)     88: (2,1,4)
    8: (4)           54: (1,2,1,2)     90: (2,1,2,2)
   10: (2,2)         58: (1,1,2,2)     91: (2,1,2,1,1)
   16: (5)           64: (7)           93: (2,1,1,2,1)
   21: (2,2,1)       69: (4,2,1)       94: (2,1,1,1,2)
   22: (2,1,2)       70: (4,1,2)       98: (1,4,2)
   26: (1,2,2)       73: (3,3,1)      100: (1,3,3)
   32: (6)           74: (3,2,2)      104: (1,2,4)
   34: (4,2)         76: (3,1,3)      106: (1,2,2,2)
   36: (3,3)         81: (2,4,1)      107: (1,2,2,1,1)
   40: (2,4)         82: (2,3,2)      109: (1,2,1,2,1)
   42: (2,2,2)       84: (2,2,3)      110: (1,2,1,1,2)
   43: (2,2,1,1)     85: (2,2,2,1)    117: (1,1,2,2,1)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !CoprimeQ@@stc[#]&]

Crossrefs

The complement is A333227.

The version without singletons is A335236.

Ignoring repeated parts gives A335238.

Singleton or pairwise coprime partitions are counted by A051424.

Singleton or pairwise coprime sets are ranked by A087087.

Numbers whose binary indices are pairwise coprime are A326675.

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.

Cf. A007360, A048793, A101268, A291166, A302569, A335235, A335237.

Sequence in context: A068382 A356232 A025612 * A102248 A161374 A272062

Adjacent sequences: A335236 A335237 A335238 * A335240 A335241 A335242

Keyword

nonn

Author

Gus Wiseman, May 28 2020

Status

approved