cp's OEIS Frontend

A337696 Numbers k such that the k-th composition in standard order (A066099) is strict and pairwise non-coprime, meaning the parts are distinct and any two of them have a common divisor > 1.

%I A337696 #16 Oct 13 2020 14:33:27
%S A337696 0,2,4,8,16,32,34,40,64,128,130,160,256,260,288,512,514,520,544,640,
%T A337696 1024,2048,2050,2052,2056,2082,2088,2176,2178,2208,2304,2560,2568,
%U A337696 2592,4096,8192,8194,8200,8224,8226,8232,8320,8704,8706,8832,10240,10248,10368
%N A337696 Numbers k such that the k-th composition in standard order (A066099) is strict and pairwise non-coprime, meaning the parts are distinct and any two of them have a common divisor > 1.
%C A337696 Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).
%C A337696 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 A337696 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A337696 Intersection of A337666 and A233564.
%e A337696 The sequence together with the corresponding compositions begins:
%e A337696        0: ()        512: (10)       2304: (3,9)
%e A337696        2: (2)       514: (8,2)      2560: (2,10)
%e A337696        4: (3)       520: (6,4)      2568: (2,6,4)
%e A337696        8: (4)       544: (4,6)      2592: (2,4,6)
%e A337696       16: (5)       640: (2,8)      4096: (13)
%e A337696       32: (6)      1024: (11)       8192: (14)
%e A337696       34: (4,2)    2048: (12)       8194: (12,2)
%e A337696       40: (2,4)    2050: (10,2)     8200: (10,4)
%e A337696       64: (7)      2052: (9,3)      8224: (8,6)
%e A337696      128: (8)      2056: (8,4)      8226: (8,4,2)
%e A337696      130: (6,2)    2082: (6,4,2)    8232: (8,2,4)
%e A337696      160: (2,6)    2088: (6,2,4)    8320: (6,8)
%e A337696      256: (9)      2176: (4,8)      8704: (4,10)
%e A337696      260: (6,3)    2178: (4,6,2)    8706: (4,8,2)
%e A337696      288: (3,6)    2208: (4,2,6)    8832: (4,2,8)
%t A337696 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A337696 stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
%t A337696 Select[Range[0,1000],UnsameQ@@stc[#]&&stabQ[stc[#],CoprimeQ]&]
%Y A337696 A318719 gives the Heinz numbers of the unordered version, with non-strict version A337694.
%Y A337696 A337667 counts the non-strict version.
%Y A337696 A337983 counts these compositions, with unordered version A318717.
%Y A337696 A051185 counts intersecting set-systems, with spanning case A305843.
%Y A337696 A200976 and A328673 count the unordered non-strict version.
%Y A337696 A337462 counts pairwise coprime compositions.
%Y A337696 A318749 counts pairwise non-coprime factorizations, with strict case A319786.
%Y A337696 All of the following pertain to compositions in standard order (A066099):
%Y A337696 - A000120 is length.
%Y A337696 - A070939 is sum.
%Y A337696 - A124767 counts runs.
%Y A337696 - A233564 ranks strict compositions.
%Y A337696 - A272919 ranks constant compositions.
%Y A337696 - A333219 is Heinz number.
%Y A337696 - A333227 ranks pairwise coprime compositions, or A335235 if singletons are considered coprime.
%Y A337696 - A333228 ranks compositions whose distinct parts are pairwise coprime.
%Y A337696 - A335236 ranks compositions neither a singleton nor pairwise coprime.
%Y A337696 - A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
%Y A337696 - A337666 ranks the non-strict version.
%Y A337696 Cf. A082024, A101268, A302797, A305713, A319752, A327040, A327516, A336737, A337599, A337604, A337605.
%K A337696 nonn
%O A337696 1,2
%A A337696 _Gus Wiseman_, Oct 06 2020