A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

0, 1, 2, 3, 4, 7, 8, 10, 11, 13, 14, 15, 16, 31, 32, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 127, 128, 136, 138, 139, 141, 142, 143, 162, 163, 168, 170, 171, 173, 174, 175, 177, 181, 182, 183, 184

Offset: 1

Author

Gus Wiseman, Nov 15 2021

Keywords

• nonn

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.

Examples

			The terms and corresponding compositions begin:
      0: ()              36: (3,3)           54: (1,2,1,2)
      1: (1)             37: (3,2,1)         55: (1,2,1,1,1)
      2: (2)             38: (3,1,2)         57: (1,1,3,1)
      3: (1,1)           39: (3,1,1,1)       58: (1,1,2,2)
      4: (3)             41: (2,3,1)         59: (1,1,2,1,1)
      7: (1,1,1)         42: (2,2,2)         60: (1,1,1,3)
      8: (4)             43: (2,2,1,1)       61: (1,1,1,2,1)
     10: (2,2)           44: (2,1,3)         62: (1,1,1,1,2)
     11: (2,1,1)         45: (2,1,2,1)       63: (1,1,1,1,1,1)
     13: (1,2,1)         46: (2,1,1,2)       64: (7)
     14: (1,1,2)         47: (2,1,1,1,1)    127: (1,1,1,1,1,1,1)
     15: (1,1,1,1)       50: (1,3,2)        128: (8)
     16: (5)             51: (1,3,1,1)      136: (4,4)
     31: (1,1,1,1,1)     52: (1,2,3)        138: (4,2,2)
     32: (6)             53: (1,2,2,1)      139: (4,2,1,1)
		

Crossrefs

Looking at length instead of parts gives A096199.

These composition are counted by A100346.

A version counting subsets instead of compositions is A125297.

An unordered version is A326841, counted by A018818.

A011782 counts compositions.

A316413 ranks partitions with sum divisible by length, counted by A067538.

A319333 ranks partitions with sum equal to lcm, counted by A074761.

Cf. A003242, A056239, A238279, A326836, A326842.

Statistics of standard compositions:

- The compositions themselves are the rows of A066099.

- Number of parts is given by A000120, distinct A334028.

- Sum and product of parts are given by A070939 and A124758.

- Maximum and minimum parts are given by A333766 and A333768.

Classes of standard compositions:

- Partitions and strict partitions are ranked by A114994 and A333256.

- Multisets and sets are ranked by A225620 and A333255.

- Strict and constant compositions are ranked by A233564 and A272919.

- Permutations are ranked by A333218.

- Relatively prime compositions are ranked by A291166*, complement A291165.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],#==0||Divisible[Total[stc[#]],LCM@@stc[#]]&]