A335236 - OEIS

%I #9 May 30 2020 09:18:48

%S 0,10,21,22,26,34,36,40,42,43,45,46,53,54,58,69,70,73,74,76,81,82,84,

%T 85,86,87,88,90,91,93,94,98,100,104,106,107,109,110,117,118,122,130,

%U 136,138,139,141,142,146,147,148,149,150,153,154,156,160,162,163,164

%N Numbers k such that the k-th composition in standard order (A066099) is not a singleton nor pairwise coprime.

%C These are compositions whose product is strictly greater than the LCM of their parts.

%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: ()            74: (3,2,2)        109: (1,2,1,2,1)

%e    10: (2,2)         76: (3,1,3)        110: (1,2,1,1,2)

%e    21: (2,2,1)       81: (2,4,1)        117: (1,1,2,2,1)

%e    22: (2,1,2)       82: (2,3,2)        118: (1,1,2,1,2)

%e    26: (1,2,2)       84: (2,2,3)        122: (1,1,1,2,2)

%e    34: (4,2)         85: (2,2,2,1)      130: (6,2)

%e    36: (3,3)         86: (2,2,1,2)      136: (4,4)

%e    40: (2,4)         87: (2,2,1,1,1)    138: (4,2,2)

%e    42: (2,2,2)       88: (2,1,4)        139: (4,2,1,1)

%e    43: (2,2,1,1)     90: (2,1,2,2)      141: (4,1,2,1)

%e    45: (2,1,2,1)     91: (2,1,2,1,1)    142: (4,1,1,2)

%e    46: (2,1,1,2)     93: (2,1,1,2,1)    146: (3,3,2)

%e    53: (1,2,2,1)     94: (2,1,1,1,2)    147: (3,3,1,1)

%e    54: (1,2,1,2)     98: (1,4,2)        148: (3,2,3)

%e    58: (1,1,2,2)    100: (1,3,3)        149: (3,2,2,1)

%e    69: (4,2,1)      104: (1,2,4)        150: (3,2,1,2)

%e    70: (4,1,2)      106: (1,2,2,2)      153: (3,1,3,1)

%e    73: (3,3,1)      107: (1,2,2,1,1)    154: (3,1,2,2)

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

%t Select[Range[0,100],!(Length[stc[#]]==1||CoprimeQ@@stc[#])&]

%Y The version for prime indices is A316438.

%Y The version for binary indices is A335237.

%Y The complement is A335235.

%Y The version with singletons allowed is A335239.

%Y Binary indices are pairwise coprime or a singleton: A087087.

%Y The version counting partitions is 1 + A335240.

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

%Y - Length is A000120.

%Y - The parts are row k of A066099.

%Y - Sum is A070939.

%Y - Product is A124758.

%Y - Reverse is A228351

%Y - GCD is A326674.

%Y - Heinz number is A333219.

%Y - LCM is A333226.

%Y Cf. A007360, A048793, A051424, A101268, A272919, A291166, A302569, A326675, A333227, A333228, A335238.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 28 2020