cp's OEIS Frontend

A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).

%I A335238 #5 May 30 2020 09:19:02
%S A335238 0,2,4,8,10,16,32,34,36,40,42,64,69,70,81,88,98,104,128,130,136,138,
%T A335238 139,141,142,160,162,163,168,170,177,184,197,198,209,216,226,232,256,
%U A335238 260,261,262,274,276,277,278,279,282,283,285,286,288,290,292,296,321
%N A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).
%C A335238 We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
%C A335238 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.
%e A335238 The sequence together with the corresponding compositions begins:
%e A335238     0: ()          88: (2,1,4)      177: (2,1,4,1)
%e A335238     2: (2)         98: (1,4,2)      184: (2,1,1,4)
%e A335238     4: (3)        104: (1,2,4)      197: (1,4,2,1)
%e A335238     8: (4)        128: (8)          198: (1,4,1,2)
%e A335238    10: (2,2)      130: (6,2)        209: (1,2,4,1)
%e A335238    16: (5)        136: (4,4)        216: (1,2,1,4)
%e A335238    32: (6)        138: (4,2,2)      226: (1,1,4,2)
%e A335238    34: (4,2)      139: (4,2,1,1)    232: (1,1,2,4)
%e A335238    36: (3,3)      141: (4,1,2,1)    256: (9)
%e A335238    40: (2,4)      142: (4,1,1,2)    260: (6,3)
%e A335238    42: (2,2,2)    160: (2,6)        261: (6,2,1)
%e A335238    64: (7)        162: (2,4,2)      262: (6,1,2)
%e A335238    69: (4,2,1)    163: (2,4,1,1)    274: (4,3,2)
%e A335238    70: (4,1,2)    168: (2,2,4)      276: (4,2,3)
%e A335238    81: (2,4,1)    170: (2,2,2,2)    277: (4,2,2,1)
%t A335238 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A335238 Select[Range[0,100],!CoprimeQ@@Union[stc[#]]&]
%Y A335238 The complement is A333228.
%Y A335238 Not ignoring repeated parts gives A335239.
%Y A335238 Singleton or pairwise coprime partitions are counted by A051424.
%Y A335238 Singleton or pairwise coprime sets are ranked by A087087.
%Y A335238 Coprime partitions are counted by A327516.
%Y A335238 Non-coprime partitions are counted by A335240.
%Y A335238 All of the following pertain to compositions in standard order (A066099):
%Y A335238 - Length is A000120.
%Y A335238 - Sum is A070939.
%Y A335238 - Product is A124758.
%Y A335238 - Reverse is A228351
%Y A335238 - GCD is A326674.
%Y A335238 - Heinz number is A333219.
%Y A335238 - LCM is A333226.
%Y A335238 - Coprime compositions are A333227.
%Y A335238 - Compositions whose distinct parts are coprime are A333228.
%Y A335238 - Number of distinct parts is A334028.
%Y A335238 Cf. A007360, A048793, A101268, A291166, A302569, A326675, A335235, A335236, A335237.
%K A335238 nonn
%O A335238 1,2
%A A335238 _Gus Wiseman_, May 28 2020