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A335235 Numbers k such that the k-th composition in standard order (A066099) is pairwise coprime, where a singleton is always considered coprime.

%I A335235 #8 May 30 2020 09:18:39
%S A335235 1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,20,23,24,25,27,28,29,30,
%T A335235 31,32,33,35,37,38,39,41,44,47,48,49,50,51,52,55,56,57,59,60,61,62,63,
%U A335235 64,65,66,67,68,71,72,75,77,78,79,80,83,89,92,95,96,97
%N A335235 Numbers k such that the k-th composition in standard order (A066099) is pairwise coprime, where a singleton is always considered coprime.
%C A335235 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 A335235 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 A335235 The sequence together with the corresponding compositions begins:
%e A335235    1: (1)          20: (2,3)          48: (1,5)
%e A335235    2: (2)          23: (2,1,1,1)      49: (1,4,1)
%e A335235    3: (1,1)        24: (1,4)          50: (1,3,2)
%e A335235    4: (3)          25: (1,3,1)        51: (1,3,1,1)
%e A335235    5: (2,1)        27: (1,2,1,1)      52: (1,2,3)
%e A335235    6: (1,2)        28: (1,1,3)        55: (1,2,1,1,1)
%e A335235    7: (1,1,1)      29: (1,1,2,1)      56: (1,1,4)
%e A335235    8: (4)          30: (1,1,1,2)      57: (1,1,3,1)
%e A335235    9: (3,1)        31: (1,1,1,1,1)    59: (1,1,2,1,1)
%e A335235   11: (2,1,1)      32: (6)            60: (1,1,1,3)
%e A335235   12: (1,3)        33: (5,1)          61: (1,1,1,2,1)
%e A335235   13: (1,2,1)      35: (4,1,1)        62: (1,1,1,1,2)
%e A335235   14: (1,1,2)      37: (3,2,1)        63: (1,1,1,1,1,1)
%e A335235   15: (1,1,1,1)    38: (3,1,2)        64: (7)
%e A335235   16: (5)          39: (3,1,1,1)      65: (6,1)
%e A335235   17: (4,1)        41: (2,3,1)        66: (5,2)
%e A335235   18: (3,2)        44: (2,1,3)        67: (5,1,1)
%e A335235   19: (3,1,1)      47: (2,1,1,1,1)    68: (4,3)
%t A335235 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A335235 Select[Range[0,100],Length[stc[#]]==1||CoprimeQ@@stc[#]&]
%Y A335235 The version counting partitions is A051424, with strict case A007360.
%Y A335235 The version for binary indices is A087087.
%Y A335235 The version counting compositions is A101268.
%Y A335235 The version for prime indices is A302569.
%Y A335235 The case without singletons is A333227.
%Y A335235 The complement is A335236.
%Y A335235 Numbers whose binary indices are pairwise coprime are A326675.
%Y A335235 Coprime partitions are counted by A327516.
%Y A335235 All of the following pertain to compositions in standard order:
%Y A335235 - Length is A000120.
%Y A335235 - The parts are row k of A066099.
%Y A335235 - Sum is A070939.
%Y A335235 - Product is A124758.
%Y A335235 - Reverse is A228351
%Y A335235 - GCD is A326674.
%Y A335235 - Heinz number is A333219.
%Y A335235 - LCM is A333226.
%Y A335235 Cf. A048793, A272919, A291166, A302569, A335236, A335237, A335238, A335239, A335240.
%K A335235 nonn
%O A335235 1,2
%A A335235 _Gus Wiseman_, May 28 2020