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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).

%I A335239 #6 May 30 2020 09:19:09
%S A335239 0,2,4,8,10,16,21,22,26,32,34,36,40,42,43,45,46,53,54,58,64,69,70,73,
%T A335239 74,76,81,82,84,85,86,87,88,90,91,93,94,98,100,104,106,107,109,110,
%U A335239 117,118,122,128,130,136,138,139,141,142,146,147,148,149,150,153
%N 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).
%C A335239 We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
%C A335239 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 A335239 The sequence together with the corresponding compositions begins:
%e A335239     0: ()            45: (2,1,2,1)     86: (2,2,1,2)
%e A335239     2: (2)           46: (2,1,1,2)     87: (2,2,1,1,1)
%e A335239     4: (3)           53: (1,2,2,1)     88: (2,1,4)
%e A335239     8: (4)           54: (1,2,1,2)     90: (2,1,2,2)
%e A335239    10: (2,2)         58: (1,1,2,2)     91: (2,1,2,1,1)
%e A335239    16: (5)           64: (7)           93: (2,1,1,2,1)
%e A335239    21: (2,2,1)       69: (4,2,1)       94: (2,1,1,1,2)
%e A335239    22: (2,1,2)       70: (4,1,2)       98: (1,4,2)
%e A335239    26: (1,2,2)       73: (3,3,1)      100: (1,3,3)
%e A335239    32: (6)           74: (3,2,2)      104: (1,2,4)
%e A335239    34: (4,2)         76: (3,1,3)      106: (1,2,2,2)
%e A335239    36: (3,3)         81: (2,4,1)      107: (1,2,2,1,1)
%e A335239    40: (2,4)         82: (2,3,2)      109: (1,2,1,2,1)
%e A335239    42: (2,2,2)       84: (2,2,3)      110: (1,2,1,1,2)
%e A335239    43: (2,2,1,1)     85: (2,2,2,1)    117: (1,1,2,2,1)
%t A335239 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A335239 Select[Range[0,100],!CoprimeQ@@stc[#]&]
%Y A335239 The complement is A333227.
%Y A335239 The version without singletons is A335236.
%Y A335239 Ignoring repeated parts gives A335238.
%Y A335239 Singleton or pairwise coprime partitions are counted by A051424.
%Y A335239 Singleton or pairwise coprime sets are ranked by A087087.
%Y A335239 Numbers whose binary indices are pairwise coprime are A326675.
%Y A335239 Coprime partitions are counted by A327516.
%Y A335239 Non-coprime partitions are counted by A335240.
%Y A335239 All of the following pertain to compositions in standard order (A066099):
%Y A335239 - Length is A000120.
%Y A335239 - Sum is A070939.
%Y A335239 - Product is A124758.
%Y A335239 - Reverse is A228351
%Y A335239 - GCD is A326674.
%Y A335239 - Heinz number is A333219.
%Y A335239 - LCM is A333226.
%Y A335239 - Coprime compositions are A333227.
%Y A335239 - Compositions whose distinct parts are coprime are A333228.
%Y A335239 - Number of distinct parts is A334028.
%Y A335239 Cf. A007360, A048793, A101268, A291166, A302569, A335235, A335237.
%K A335239 nonn
%O A335239 1,2
%A A335239 _Gus Wiseman_, May 28 2020