A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

34, 40, 69, 70, 81, 88, 98, 104, 130, 138, 139, 141, 142, 160, 162, 163, 168, 177, 184, 197, 198, 209, 216, 226, 232, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 296, 321, 324, 325, 326, 327, 328, 337, 344, 352, 354, 355, 360, 369

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..53.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Example

The sequence together with the corresponding compositions begins:
     34: (4,2)        163: (2,4,1,1)    277: (4,2,2,1)
     40: (2,4)        168: (2,2,4)      278: (4,2,1,2)
     69: (4,2,1)      177: (2,1,4,1)    279: (4,2,1,1,1)
     70: (4,1,2)      184: (2,1,1,4)    282: (4,1,2,2)
     81: (2,4,1)      197: (1,4,2,1)    283: (4,1,2,1,1)
     88: (2,1,4)      198: (1,4,1,2)    285: (4,1,1,2,1)
     98: (1,4,2)      209: (1,2,4,1)    286: (4,1,1,1,2)
    104: (1,2,4)      216: (1,2,1,4)    288: (3,6)
    130: (6,2)        226: (1,1,4,2)    290: (3,4,2)
    138: (4,2,2)      232: (1,1,2,4)    296: (3,2,4)
    139: (4,2,1,1)    260: (6,3)        321: (2,6,1)
    141: (4,1,2,1)    261: (6,2,1)      324: (2,4,3)
    142: (4,1,1,2)    262: (6,1,2)      325: (2,4,2,1)
    160: (2,6)        274: (4,3,2)      326: (2,4,1,2)
    162: (2,4,2)      276: (4,2,3)      327: (2,4,1,1,1)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !(SameQ@@stc[#]||CoprimeQ@@Union[stc[#]])&]

Crossrefs

A304712 counts the complement, with ordered version A337664.

A333228 ranks compositions whose distinct parts are pairwise coprime.

A335238 does not consider a singleton coprime unless it is (1).

A337600 counts 3-part partitions in the complement.

A000740 counts relatively prime compositions.

A051424 counts pairwise coprime or singleton partitions.

A101268 counts pairwise coprime or singleton compositions.

A327516 counts pairwise coprime partitions.

A333227 ranks pairwise coprime compositions.

A337461 counts pairwise coprime 3-part compositions.

A337561 counts pairwise coprime strict compositions.

A337665 counts compositions whose distinct parts are pairwise coprime.

A337666 ranks pairwise non-coprime compositions.

Cf. A007359, A007360, A087087, A302569, A302696, A304709, A305713, A335235, A337562, A337601, A337602, A337603.

Sequence in context: A295750 A141699 A296096 * A045044 A108303 A062695

Adjacent sequences: A337692 A337693 A337694 * A337696 A337697 A337698

Keyword

nonn

Author

Gus Wiseman, Sep 22 2020

Status

approved