%I #10 May 30 2020 19:14:25
%S 1,3,5,6,7,9,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30,
%T 31,33,35,37,38,39,41,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,
%U 59,60,61,62,63,65,66,67,68,71,72,73,74,75,76,77,78,79,80
%N Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).
%C First differs from A291166 in lacking 69, which corresponds to the composition (4,2,1).
%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.
%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) 21: (2,2,1) 39: (3,1,1,1)
%e 3: (1,1) 22: (2,1,2) 41: (2,3,1)
%e 5: (2,1) 23: (2,1,1,1) 43: (2,2,1,1)
%e 6: (1,2) 24: (1,4) 44: (2,1,3)
%e 7: (1,1,1) 25: (1,3,1) 45: (2,1,2,1)
%e 9: (3,1) 26: (1,2,2) 46: (2,1,1,2)
%e 11: (2,1,1) 27: (1,2,1,1) 47: (2,1,1,1,1)
%e 12: (1,3) 28: (1,1,3) 48: (1,5)
%e 13: (1,2,1) 29: (1,1,2,1) 49: (1,4,1)
%e 14: (1,1,2) 30: (1,1,1,2) 50: (1,3,2)
%e 15: (1,1,1,1) 31: (1,1,1,1,1) 51: (1,3,1,1)
%e 17: (4,1) 33: (5,1) 52: (1,2,3)
%e 18: (3,2) 35: (4,1,1) 53: (1,2,2,1)
%e 19: (3,1,1) 37: (3,2,1) 54: (1,2,1,2)
%e 20: (2,3) 38: (3,1,2) 55: (1,2,1,1,1)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Select[Range[0,120],CoprimeQ@@Union[stc[#]]&]
%Y Pairwise coprime or singleton partitions are A051424.
%Y Coprime or singleton sets are ranked by A087087.
%Y The version for relatively prime instead of coprime appears to be A291166.
%Y Numbers whose binary indices are pairwise coprime are A326675.
%Y Coprime partitions are counted by A327516.
%Y Not ignoring repeated parts gives A333227.
%Y The complement is A335238.
%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 - Number of distinct parts is A334028.
%Y Cf. A007360, A048793, A101268, A233564, A272919, A291166, A302569, A335235, A335236, A335237, A335239.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 28 2020