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