A335236 Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.

0, 10, 21, 22, 26, 34, 36, 40, 42, 43, 45, 46, 53, 54, 58, 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, 130, 136, 138, 139, 141, 142, 146, 147, 148, 149, 150, 153, 154, 156, 160, 162, 163, 164

Offset: 1

Author

Gus Wiseman, May 28 2020

Keywords

• nonn

Comments

These are compositions whose product is strictly greater than the LCM of their parts.

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:
    0: ()            74: (3,2,2)        109: (1,2,1,2,1)
   10: (2,2)         76: (3,1,3)        110: (1,2,1,1,2)
   21: (2,2,1)       81: (2,4,1)        117: (1,1,2,2,1)
   22: (2,1,2)       82: (2,3,2)        118: (1,1,2,1,2)
   26: (1,2,2)       84: (2,2,3)        122: (1,1,1,2,2)
   34: (4,2)         85: (2,2,2,1)      130: (6,2)
   36: (3,3)         86: (2,2,1,2)      136: (4,4)
   40: (2,4)         87: (2,2,1,1,1)    138: (4,2,2)
   42: (2,2,2)       88: (2,1,4)        139: (4,2,1,1)
   43: (2,2,1,1)     90: (2,1,2,2)      141: (4,1,2,1)
   45: (2,1,2,1)     91: (2,1,2,1,1)    142: (4,1,1,2)
   46: (2,1,1,2)     93: (2,1,1,2,1)    146: (3,3,2)
   53: (1,2,2,1)     94: (2,1,1,1,2)    147: (3,3,1,1)
   54: (1,2,1,2)     98: (1,4,2)        148: (3,2,3)
   58: (1,1,2,2)    100: (1,3,3)        149: (3,2,2,1)
   69: (4,2,1)      104: (1,2,4)        150: (3,2,1,2)
   70: (4,1,2)      106: (1,2,2,2)      153: (3,1,3,1)
   73: (3,3,1)      107: (1,2,2,1,1)    154: (3,1,2,2)
		

Links

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

Crossrefs

The version for prime indices is A316438.

The version for binary indices is A335237.

The complement is A335235.

The version with singletons allowed is A335239.

Binary indices are pairwise coprime or a singleton: A087087.

The version counting partitions is 1 + A335240.

All of the following pertain to compositions in standard order:

- Length is A000120.

- The parts are row k of A066099.

- Sum is A070939.

- Product is A124758.

- Reverse is A228351

- GCD is A326674.

- Heinz number is A333219.

- LCM is A333226.

Cf. A007360, A048793, A051424, A101268, A272919, A291166, A302569, A326675, A333227, A333228, A335238.

Programs

Mathematica

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