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A333228 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).

%I A333228 #10 May 30 2020 19:14:25
%S A333228 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 A333228 31,33,35,37,38,39,41,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,
%U A333228 59,60,61,62,63,65,66,67,68,71,72,73,74,75,76,77,78,79,80
%N A333228 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 A333228 First differs from A291166 in lacking 69, which corresponds to the composition (4,2,1).
%C A333228 We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
%C A333228 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 A333228 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 A333228 The sequence together with the corresponding compositions begins:
%e A333228    1: (1)          21: (2,2,1)        39: (3,1,1,1)
%e A333228    3: (1,1)        22: (2,1,2)        41: (2,3,1)
%e A333228    5: (2,1)        23: (2,1,1,1)      43: (2,2,1,1)
%e A333228    6: (1,2)        24: (1,4)          44: (2,1,3)
%e A333228    7: (1,1,1)      25: (1,3,1)        45: (2,1,2,1)
%e A333228    9: (3,1)        26: (1,2,2)        46: (2,1,1,2)
%e A333228   11: (2,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
%e A333228   12: (1,3)        28: (1,1,3)        48: (1,5)
%e A333228   13: (1,2,1)      29: (1,1,2,1)      49: (1,4,1)
%e A333228   14: (1,1,2)      30: (1,1,1,2)      50: (1,3,2)
%e A333228   15: (1,1,1,1)    31: (1,1,1,1,1)    51: (1,3,1,1)
%e A333228   17: (4,1)        33: (5,1)          52: (1,2,3)
%e A333228   18: (3,2)        35: (4,1,1)        53: (1,2,2,1)
%e A333228   19: (3,1,1)      37: (3,2,1)        54: (1,2,1,2)
%e A333228   20: (2,3)        38: (3,1,2)        55: (1,2,1,1,1)
%t A333228 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333228 Select[Range[0,120],CoprimeQ@@Union[stc[#]]&]
%Y A333228 Pairwise coprime or singleton partitions are A051424.
%Y A333228 Coprime or singleton sets are ranked by A087087.
%Y A333228 The version for relatively prime instead of coprime appears to be A291166.
%Y A333228 Numbers whose binary indices are pairwise coprime are A326675.
%Y A333228 Coprime partitions are counted by A327516.
%Y A333228 Not ignoring repeated parts gives A333227.
%Y A333228 The complement is A335238.
%Y A333228 All of the following pertain to compositions in standard order (A066099):
%Y A333228 - Length is A000120.
%Y A333228 - Sum is A070939.
%Y A333228 - Product is A124758.
%Y A333228 - Reverse is A228351
%Y A333228 - GCD is A326674.
%Y A333228 - Heinz number is A333219.
%Y A333228 - LCM is A333226.
%Y A333228 - Number of distinct parts is A334028.
%Y A333228 Cf. A007360, A048793, A101268, A233564, A272919, A291166, A302569, A335235, A335236, A335237, A335239.
%K A333228 nonn
%O A333228 1,2
%A A333228 _Gus Wiseman_, May 28 2020