A333227 - OEIS

%I #7 Mar 12 2022 14:09:25

%S 1,3,5,6,7,9,11,12,13,14,15,17,18,19,20,23,24,25,27,28,29,30,31,33,35,

%T 37,38,39,41,44,47,48,49,50,51,52,55,56,57,59,60,61,62,63,65,66,67,68,

%U 71,72,75,77,78,79,80,83,89,92,95,96,97,99,101,102,103,105

%N Numbers k such that the k-th composition in standard order is pairwise coprime, where a singleton is not coprime unless it is (1).

%C This is the definition used for CoprimeQ in Mathematica.

%C The k-th composition in standard order (row k of 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.

%e The sequence together with the corresponding compositions begins:

%e    1: (1)          27: (1,2,1,1)      55: (1,2,1,1,1)

%e    3: (1,1)        28: (1,1,3)        56: (1,1,4)

%e    5: (2,1)        29: (1,1,2,1)      57: (1,1,3,1)

%e    6: (1,2)        30: (1,1,1,2)      59: (1,1,2,1,1)

%e    7: (1,1,1)      31: (1,1,1,1,1)    60: (1,1,1,3)

%e    9: (3,1)        33: (5,1)          61: (1,1,1,2,1)

%e   11: (2,1,1)      35: (4,1,1)        62: (1,1,1,1,2)

%e   12: (1,3)        37: (3,2,1)        63: (1,1,1,1,1,1)

%e   13: (1,2,1)      38: (3,1,2)        65: (6,1)

%e   14: (1,1,2)      39: (3,1,1,1)      66: (5,2)

%e   15: (1,1,1,1)    41: (2,3,1)        67: (5,1,1)

%e   17: (4,1)        44: (2,1,3)        68: (4,3)

%e   18: (3,2)        47: (2,1,1,1,1)    71: (4,1,1,1)

%e   19: (3,1,1)      48: (1,5)          72: (3,4)

%e   20: (2,3)        49: (1,4,1)        75: (3,2,1,1)

%e   23: (2,1,1,1)    50: (1,3,2)        77: (3,1,2,1)

%e   24: (1,4)        51: (1,3,1,1)      78: (3,1,1,2)

%e   25: (1,3,1)      52: (1,2,3)        79: (3,1,1,1,1)

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

%t Select[Range[0,120],CoprimeQ@@stc[#]&]

%Y A different ranking of the same compositions is A326675.

%Y Ignoring repeated parts gives A333228.

%Y Let q(k) be the k-th composition in standard order:

%Y - The terms of q(k) are row k of A066099.

%Y - The sum of q(k) is A070939(k).

%Y - The product of q(k) is A124758(k).

%Y - q(k) has A124767(k) runs and A333381(k) anti-runs.

%Y - The GCD of q(k) is A326674(k).

%Y - The Heinz number of q(k) is A333219(k).

%Y - The LCM of q(k) is A333226(k).

%Y Coprime or singleton sets are ranked by A087087.

%Y Strict compositions are ranked by A233564.

%Y Constant compositions are ranked by A272919.

%Y Relatively prime compositions appear to be ranked by A291166.

%Y Normal compositions are ranked by A333217.

%Y Cf. A000120, A029931, A048793, A096111, A114994, A225620, A228351.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 27 2020