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Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.
13

%I #10 Oct 13 2020 14:33:04

%S 0,2,4,8,10,16,32,34,36,40,42,64,128,130,136,138,160,162,168,170,256,

%T 260,288,292,512,514,520,522,528,544,546,552,554,640,642,648,650,672,

%U 674,680,682,1024,2048,2050,2052,2056,2058,2080,2082,2084,2088,2090,2176

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

%C Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).

%C This is a ranking sequence for pairwise non-coprime compositions.

%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 0: () 138: (4,2,2) 546: (4,4,2)

%e 2: (2) 160: (2,6) 552: (4,2,4)

%e 4: (3) 162: (2,4,2) 554: (4,2,2,2)

%e 8: (4) 168: (2,2,4) 640: (2,8)

%e 10: (2,2) 170: (2,2,2,2) 642: (2,6,2)

%e 16: (5) 256: (9) 648: (2,4,4)

%e 32: (6) 260: (6,3) 650: (2,4,2,2)

%e 34: (4,2) 288: (3,6) 672: (2,2,6)

%e 36: (3,3) 292: (3,3,3) 674: (2,2,4,2)

%e 40: (2,4) 512: (10) 680: (2,2,2,4)

%e 42: (2,2,2) 514: (8,2) 682: (2,2,2,2,2)

%e 64: (7) 520: (6,4) 1024: (11)

%e 128: (8) 522: (6,2,2) 2048: (12)

%e 130: (6,2) 528: (5,5) 2050: (10,2)

%e 136: (4,4) 544: (4,6) 2052: (9,3)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];

%t Select[Range[0,1000],stabQ[stc[#],CoprimeQ]&]

%Y A337604 counts these compositions of length 3.

%Y A337667 counts these compositions.

%Y A337694 is the version for Heinz numbers of partitions.

%Y A337696 is the strict case.

%Y A051185 and A305843 (covering) count pairwise intersecting set-systems.

%Y A101268 counts pairwise coprime or singleton compositions.

%Y A200976 and A328673 count pairwise non-coprime partitions.

%Y A318717 counts strict pairwise non-coprime partitions.

%Y A327516 counts pairwise coprime partitions.

%Y A335236 ranks compositions neither a singleton nor pairwise coprime.

%Y A337462 counts pairwise coprime compositions.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - A000120 is length.

%Y - A070939 is sum.

%Y - A124767 counts runs.

%Y - A233564 ranks strict compositions.

%Y - A272919 ranks constant compositions.

%Y - A291166 appears to rank relatively prime compositions.

%Y - A326674 is greatest common divisor.

%Y - A333219 is Heinz number.

%Y - A333227 ranks coprime (Mathematica definition) compositions.

%Y - A333228 ranks compositions with distinct parts coprime.

%Y - A335235 ranks singleton or coprime compositions.

%Y Cf. A082024, A284825, A305713, A319752, A319786, A327039, A327040, A336737, A337599, A337605.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 05 2020