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A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

%I A337666 #10 Oct 13 2020 14:33:04
%S A337666 0,2,4,8,10,16,32,34,36,40,42,64,128,130,136,138,160,162,168,170,256,
%T A337666 260,288,292,512,514,520,522,528,544,546,552,554,640,642,648,650,672,
%U A337666 674,680,682,1024,2048,2050,2052,2056,2058,2080,2082,2084,2088,2090,2176
%N A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.
%C A337666 Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).
%C A337666 This is a ranking sequence for pairwise non-coprime compositions.
%C A337666 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 A337666 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 A337666 The sequence together with the corresponding compositions begins:
%e A337666        0: ()          138: (4,2,2)       546: (4,4,2)
%e A337666        2: (2)         160: (2,6)         552: (4,2,4)
%e A337666        4: (3)         162: (2,4,2)       554: (4,2,2,2)
%e A337666        8: (4)         168: (2,2,4)       640: (2,8)
%e A337666       10: (2,2)       170: (2,2,2,2)     642: (2,6,2)
%e A337666       16: (5)         256: (9)           648: (2,4,4)
%e A337666       32: (6)         260: (6,3)         650: (2,4,2,2)
%e A337666       34: (4,2)       288: (3,6)         672: (2,2,6)
%e A337666       36: (3,3)       292: (3,3,3)       674: (2,2,4,2)
%e A337666       40: (2,4)       512: (10)          680: (2,2,2,4)
%e A337666       42: (2,2,2)     514: (8,2)         682: (2,2,2,2,2)
%e A337666       64: (7)         520: (6,4)        1024: (11)
%e A337666      128: (8)         522: (6,2,2)      2048: (12)
%e A337666      130: (6,2)       528: (5,5)        2050: (10,2)
%e A337666      136: (4,4)       544: (4,6)        2052: (9,3)
%t A337666 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A337666 stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
%t A337666 Select[Range[0,1000],stabQ[stc[#],CoprimeQ]&]
%Y A337666 A337604 counts these compositions of length 3.
%Y A337666 A337667 counts these compositions.
%Y A337666 A337694 is the version for Heinz numbers of partitions.
%Y A337666 A337696 is the strict case.
%Y A337666 A051185 and A305843 (covering) count pairwise intersecting set-systems.
%Y A337666 A101268 counts pairwise coprime or singleton compositions.
%Y A337666 A200976 and A328673 count pairwise non-coprime partitions.
%Y A337666 A318717 counts strict pairwise non-coprime partitions.
%Y A337666 A327516 counts pairwise coprime partitions.
%Y A337666 A335236 ranks compositions neither a singleton nor pairwise coprime.
%Y A337666 A337462 counts pairwise coprime compositions.
%Y A337666 All of the following pertain to compositions in standard order (A066099):
%Y A337666 - A000120 is length.
%Y A337666 - A070939 is sum.
%Y A337666 - A124767 counts runs.
%Y A337666 - A233564 ranks strict compositions.
%Y A337666 - A272919 ranks constant compositions.
%Y A337666 - A291166 appears to rank relatively prime compositions.
%Y A337666 - A326674 is greatest common divisor.
%Y A337666 - A333219 is Heinz number.
%Y A337666 - A333227 ranks coprime (Mathematica definition) compositions.
%Y A337666 - A333228 ranks compositions with distinct parts coprime.
%Y A337666 - A335235 ranks singleton or coprime compositions.
%Y A337666 Cf. A082024, A284825, A305713, A319752, A319786, A327039, A327040, A336737, A337599, A337605.
%K A337666 nonn
%O A337666 1,2
%A A337666 _Gus Wiseman_, Oct 05 2020