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