A335238 - OEIS

%I #5 May 30 2020 09:19:02

%S 0,2,4,8,10,16,32,34,36,40,42,64,69,70,81,88,98,104,128,130,136,138,

%T 139,141,142,160,162,163,168,170,177,184,197,198,209,216,226,232,256,

%U 260,261,262,274,276,277,278,279,282,283,285,286,288,290,292,296,321

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

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

%C 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.

%e The sequence together with the corresponding compositions begins:

%e     0: ()          88: (2,1,4)      177: (2,1,4,1)

%e     2: (2)         98: (1,4,2)      184: (2,1,1,4)

%e     4: (3)        104: (1,2,4)      197: (1,4,2,1)

%e     8: (4)        128: (8)          198: (1,4,1,2)

%e    10: (2,2)      130: (6,2)        209: (1,2,4,1)

%e    16: (5)        136: (4,4)        216: (1,2,1,4)

%e    32: (6)        138: (4,2,2)      226: (1,1,4,2)

%e    34: (4,2)      139: (4,2,1,1)    232: (1,1,2,4)

%e    36: (3,3)      141: (4,1,2,1)    256: (9)

%e    40: (2,4)      142: (4,1,1,2)    260: (6,3)

%e    42: (2,2,2)    160: (2,6)        261: (6,2,1)

%e    64: (7)        162: (2,4,2)      262: (6,1,2)

%e    69: (4,2,1)    163: (2,4,1,1)    274: (4,3,2)

%e    70: (4,1,2)    168: (2,2,4)      276: (4,2,3)

%e    81: (2,4,1)    170: (2,2,2,2)    277: (4,2,2,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,100],!CoprimeQ@@Union[stc[#]]&]

%Y The complement is A333228.

%Y Not ignoring repeated parts gives A335239.

%Y Singleton or pairwise coprime partitions are counted by A051424.

%Y Singleton or pairwise coprime sets are ranked by A087087.

%Y Coprime partitions are counted by A327516.

%Y Non-coprime partitions are counted by A335240.

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

%Y - Length is A000120.

%Y - Sum is A070939.

%Y - Product is A124758.

%Y - Reverse is A228351

%Y - GCD is A326674.

%Y - Heinz number is A333219.

%Y - LCM is A333226.

%Y - Coprime compositions are A333227.

%Y - Compositions whose distinct parts are coprime are A333228.

%Y - Number of distinct parts is A334028.

%Y Cf. A007360, A048793, A101268, A291166, A302569, A326675, A335235, A335236, A335237.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 28 2020