A337696 Numbers k such that the k-th composition in standard order (A066099) is strict and pairwise non-coprime, meaning the parts are distinct and any two of them have a common divisor > 1.

0, 2, 4, 8, 16, 32, 34, 40, 64, 128, 130, 160, 256, 260, 288, 512, 514, 520, 544, 640, 1024, 2048, 2050, 2052, 2056, 2082, 2088, 2176, 2178, 2208, 2304, 2560, 2568, 2592, 4096, 8192, 8194, 8200, 8224, 8226, 8232, 8320, 8704, 8706, 8832, 10240, 10248, 10368

Offset: 1

Author

Gus Wiseman, Oct 06 2020

Keywords

• nonn

Comments

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

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.

Examples

			The sequence together with the corresponding compositions begins:
       0: ()        512: (10)       2304: (3,9)
       2: (2)       514: (8,2)      2560: (2,10)
       4: (3)       520: (6,4)      2568: (2,6,4)
       8: (4)       544: (4,6)      2592: (2,4,6)
      16: (5)       640: (2,8)      4096: (13)
      32: (6)      1024: (11)       8192: (14)
      34: (4,2)    2048: (12)       8194: (12,2)
      40: (2,4)    2050: (10,2)     8200: (10,4)
      64: (7)      2052: (9,3)      8224: (8,6)
     128: (8)      2056: (8,4)      8226: (8,4,2)
     130: (6,2)    2082: (6,4,2)    8232: (8,2,4)
     160: (2,6)    2088: (6,2,4)    8320: (6,8)
     256: (9)      2176: (4,8)      8704: (4,10)
     260: (6,3)    2178: (4,6,2)    8706: (4,8,2)
     288: (3,6)    2208: (4,2,6)    8832: (4,2,8)
		

Links

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

Crossrefs

A318719 gives the Heinz numbers of the unordered version, with non-strict version A337694.

A337667 counts the non-strict version.

A337983 counts these compositions, with unordered version A318717.

A051185 counts intersecting set-systems, with spanning case A305843.

A200976 and A328673 count the unordered non-strict version.

A337462 counts pairwise coprime compositions.

A318749 counts pairwise non-coprime factorizations, with strict case A319786.

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.

- A333219 is Heinz number.

- A333227 ranks pairwise coprime compositions, or A335235 if singletons are considered coprime.

- A333228 ranks compositions whose distinct parts are pairwise coprime.

- A335236 ranks compositions neither a singleton nor pairwise coprime.

- A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.

- A337666 ranks the non-strict version.

Cf. A082024, A101268, A302797, A305713, A319752, A327040, A327516, A336737, A337599, A337604, A337605.

Programs

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],UnsameQ@@stc[#]&&stabQ[stc[#],CoprimeQ]&]

Formula

Intersection of A337666 and A233564.