A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 128, 130, 136, 138, 160, 162, 168, 170, 256, 260, 288, 292, 512, 514, 520, 522, 528, 544, 546, 552, 554, 640, 642, 648, 650, 672, 674, 680, 682, 1024, 2048, 2050, 2052, 2056, 2058, 2080, 2082, 2084, 2088, 2090, 2176

Offset

1,2

Comments

Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).

This is a ranking sequence for pairwise non-coprime compositions.

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:
       0: ()          138: (4,2,2)       546: (4,4,2)
       2: (2)         160: (2,6)         552: (4,2,4)
       4: (3)         162: (2,4,2)       554: (4,2,2,2)
       8: (4)         168: (2,2,4)       640: (2,8)
      10: (2,2)       170: (2,2,2,2)     642: (2,6,2)
      16: (5)         256: (9)           648: (2,4,4)
      32: (6)         260: (6,3)         650: (2,4,2,2)
      34: (4,2)       288: (3,6)         672: (2,2,6)
      36: (3,3)       292: (3,3,3)       674: (2,2,4,2)
      40: (2,4)       512: (10)          680: (2,2,2,4)
      42: (2,2,2)     514: (8,2)         682: (2,2,2,2,2)
      64: (7)         520: (6,4)        1024: (11)
     128: (8)         522: (6,2,2)      2048: (12)
     130: (6,2)       528: (5,5)        2050: (10,2)
     136: (4,4)       544: (4,6)        2052: (9,3)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
stabQ[u_, Q_]:=And@@Not/@Q@@@Tuples[u, 2];
Select[Range[0, 1000], stabQ[stc[#], CoprimeQ]&]

Crossrefs

A337604 counts these compositions of length 3.

A337667 counts these compositions.

A337694 is the version for Heinz numbers of partitions.

A337696 is the strict case.

A051185 and A305843 (covering) count pairwise intersecting set-systems.

A101268 counts pairwise coprime or singleton compositions.

A200976 and A328673 count pairwise non-coprime partitions.

A318717 counts strict pairwise non-coprime partitions.

A327516 counts pairwise coprime partitions.

A335236 ranks compositions neither a singleton nor pairwise coprime.

A337462 counts pairwise coprime compositions.

All of the following pertain to compositions in standard order (A066099):

- A000120 is length.

- A070939 is sum.

- A124767 counts runs.

- A233564 ranks strict compositions.

- A272919 ranks constant compositions.

- A291166 appears to rank relatively prime compositions.

- A326674 is greatest common divisor.

- A333219 is Heinz number.

- A333227 ranks coprime (Mathematica definition) compositions.

- A333228 ranks compositions with distinct parts coprime.

- A335235 ranks singleton or coprime compositions.

Cf. A082024, A284825, A305713, A319752, A319786, A327039, A327040, A336737, A337599, A337605.

Sequence in context: A226816 A371291 A335238 * A291165 A083655 A335404

Adjacent sequences: A337663 A337664 A337665 * A337667 A337668 A337669

Keyword

nonn

Author

Gus Wiseman, Oct 05 2020

Status

approved