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

%I A335236 #9 May 30 2020 09:18:48
%S A335236 0,10,21,22,26,34,36,40,42,43,45,46,53,54,58,69,70,73,74,76,81,82,84,
%T A335236 85,86,87,88,90,91,93,94,98,100,104,106,107,109,110,117,118,122,130,
%U A335236 136,138,139,141,142,146,147,148,149,150,153,154,156,160,162,163,164
%N A335236 Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.
%C A335236 These are compositions whose product is strictly greater than the LCM of their parts.
%C A335236 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.
%H A335236 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A335236 The sequence together with the corresponding compositions begins:
%e A335236     0: ()            74: (3,2,2)        109: (1,2,1,2,1)
%e A335236    10: (2,2)         76: (3,1,3)        110: (1,2,1,1,2)
%e A335236    21: (2,2,1)       81: (2,4,1)        117: (1,1,2,2,1)
%e A335236    22: (2,1,2)       82: (2,3,2)        118: (1,1,2,1,2)
%e A335236    26: (1,2,2)       84: (2,2,3)        122: (1,1,1,2,2)
%e A335236    34: (4,2)         85: (2,2,2,1)      130: (6,2)
%e A335236    36: (3,3)         86: (2,2,1,2)      136: (4,4)
%e A335236    40: (2,4)         87: (2,2,1,1,1)    138: (4,2,2)
%e A335236    42: (2,2,2)       88: (2,1,4)        139: (4,2,1,1)
%e A335236    43: (2,2,1,1)     90: (2,1,2,2)      141: (4,1,2,1)
%e A335236    45: (2,1,2,1)     91: (2,1,2,1,1)    142: (4,1,1,2)
%e A335236    46: (2,1,1,2)     93: (2,1,1,2,1)    146: (3,3,2)
%e A335236    53: (1,2,2,1)     94: (2,1,1,1,2)    147: (3,3,1,1)
%e A335236    54: (1,2,1,2)     98: (1,4,2)        148: (3,2,3)
%e A335236    58: (1,1,2,2)    100: (1,3,3)        149: (3,2,2,1)
%e A335236    69: (4,2,1)      104: (1,2,4)        150: (3,2,1,2)
%e A335236    70: (4,1,2)      106: (1,2,2,2)      153: (3,1,3,1)
%e A335236    73: (3,3,1)      107: (1,2,2,1,1)    154: (3,1,2,2)
%t A335236 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A335236 Select[Range[0,100],!(Length[stc[#]]==1||CoprimeQ@@stc[#])&]
%Y A335236 The version for prime indices is A316438.
%Y A335236 The version for binary indices is A335237.
%Y A335236 The complement is A335235.
%Y A335236 The version with singletons allowed is A335239.
%Y A335236 Binary indices are pairwise coprime or a singleton: A087087.
%Y A335236 The version counting partitions is 1 + A335240.
%Y A335236 All of the following pertain to compositions in standard order:
%Y A335236 - Length is A000120.
%Y A335236 - The parts are row k of A066099.
%Y A335236 - Sum is A070939.
%Y A335236 - Product is A124758.
%Y A335236 - Reverse is A228351
%Y A335236 - GCD is A326674.
%Y A335236 - Heinz number is A333219.
%Y A335236 - LCM is A333226.
%Y A335236 Cf. A007360, A048793, A051424, A101268, A272919, A291166, A302569, A326675, A333227, A333228, A335238.
%K A335236 nonn
%O A335236 1,2
%A A335236 _Gus Wiseman_, May 28 2020