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A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

%I A337695 #10 Oct 05 2020 01:01:53
%S A337695 34,40,69,70,81,88,98,104,130,138,139,141,142,160,162,163,168,177,184,
%T A337695 197,198,209,216,226,232,260,261,262,274,276,277,278,279,282,283,285,
%U A337695 286,288,290,296,321,324,325,326,327,328,337,344,352,354,355,360,369
%N A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.
%C A337695 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 A337695 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 A337695 The sequence together with the corresponding compositions begins:
%e A337695      34: (4,2)        163: (2,4,1,1)    277: (4,2,2,1)
%e A337695      40: (2,4)        168: (2,2,4)      278: (4,2,1,2)
%e A337695      69: (4,2,1)      177: (2,1,4,1)    279: (4,2,1,1,1)
%e A337695      70: (4,1,2)      184: (2,1,1,4)    282: (4,1,2,2)
%e A337695      81: (2,4,1)      197: (1,4,2,1)    283: (4,1,2,1,1)
%e A337695      88: (2,1,4)      198: (1,4,1,2)    285: (4,1,1,2,1)
%e A337695      98: (1,4,2)      209: (1,2,4,1)    286: (4,1,1,1,2)
%e A337695     104: (1,2,4)      216: (1,2,1,4)    288: (3,6)
%e A337695     130: (6,2)        226: (1,1,4,2)    290: (3,4,2)
%e A337695     138: (4,2,2)      232: (1,1,2,4)    296: (3,2,4)
%e A337695     139: (4,2,1,1)    260: (6,3)        321: (2,6,1)
%e A337695     141: (4,1,2,1)    261: (6,2,1)      324: (2,4,3)
%e A337695     142: (4,1,1,2)    262: (6,1,2)      325: (2,4,2,1)
%e A337695     160: (2,6)        274: (4,3,2)      326: (2,4,1,2)
%e A337695     162: (2,4,2)      276: (4,2,3)      327: (2,4,1,1,1)
%t A337695 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A337695 Select[Range[0,100],!(SameQ@@stc[#]||CoprimeQ@@Union[stc[#]])&]
%Y A337695 A304712 counts the complement, with ordered version A337664.
%Y A337695 A333228 ranks compositions whose distinct parts are pairwise coprime.
%Y A337695 A335238 does not consider a singleton coprime unless it is (1).
%Y A337695 A337600 counts 3-part partitions in the complement.
%Y A337695 A000740 counts relatively prime compositions.
%Y A337695 A051424 counts pairwise coprime or singleton partitions.
%Y A337695 A101268 counts pairwise coprime or singleton compositions.
%Y A337695 A327516 counts pairwise coprime partitions.
%Y A337695 A333227 ranks pairwise coprime compositions.
%Y A337695 A337461 counts pairwise coprime 3-part compositions.
%Y A337695 A337561 counts pairwise coprime strict compositions.
%Y A337695 A337665 counts compositions whose distinct parts are pairwise coprime.
%Y A337695 A337666 ranks pairwise non-coprime compositions.
%Y A337695 Cf. A007359, A007360, A087087, A302569, A302696, A304709, A305713, A335235, A337562, A337601, A337602, A337603.
%K A337695 nonn
%O A337695 1,1
%A A337695 _Gus Wiseman_, Sep 22 2020