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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.

%I A349152 #9 Nov 22 2021 08:08:28
%S A349152 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,
%T A349152 47,50,51,52,53,54,55,57,58,59,60,61,62,63,64,127,128,136,138,139,141,
%U A349152 142,143,162,163,168,170,171,173,174,175,177,181,182,183,184
%N 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.
%C A349152 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.
%e A349152 The terms and corresponding compositions begin:
%e A349152       0: ()              36: (3,3)           54: (1,2,1,2)
%e A349152       1: (1)             37: (3,2,1)         55: (1,2,1,1,1)
%e A349152       2: (2)             38: (3,1,2)         57: (1,1,3,1)
%e A349152       3: (1,1)           39: (3,1,1,1)       58: (1,1,2,2)
%e A349152       4: (3)             41: (2,3,1)         59: (1,1,2,1,1)
%e A349152       7: (1,1,1)         42: (2,2,2)         60: (1,1,1,3)
%e A349152       8: (4)             43: (2,2,1,1)       61: (1,1,1,2,1)
%e A349152      10: (2,2)           44: (2,1,3)         62: (1,1,1,1,2)
%e A349152      11: (2,1,1)         45: (2,1,2,1)       63: (1,1,1,1,1,1)
%e A349152      13: (1,2,1)         46: (2,1,1,2)       64: (7)
%e A349152      14: (1,1,2)         47: (2,1,1,1,1)    127: (1,1,1,1,1,1,1)
%e A349152      15: (1,1,1,1)       50: (1,3,2)        128: (8)
%e A349152      16: (5)             51: (1,3,1,1)      136: (4,4)
%e A349152      31: (1,1,1,1,1)     52: (1,2,3)        138: (4,2,2)
%e A349152      32: (6)             53: (1,2,2,1)      139: (4,2,1,1)
%t A349152 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A349152 Select[Range[0,100],#==0||Divisible[Total[stc[#]],LCM@@stc[#]]&]
%Y A349152 Looking at length instead of parts gives A096199.
%Y A349152 These composition are counted by A100346.
%Y A349152 A version counting subsets instead of compositions is A125297.
%Y A349152 An unordered version is A326841, counted by A018818.
%Y A349152 A011782 counts compositions.
%Y A349152 A316413 ranks partitions with sum divisible by length, counted by A067538.
%Y A349152 A319333 ranks partitions with sum equal to lcm, counted by A074761.
%Y A349152 Cf. A003242, A056239, A238279, A326836, A326842.
%Y A349152 Statistics of standard compositions:
%Y A349152 - The compositions themselves are the rows of A066099.
%Y A349152 - Number of parts is given by A000120, distinct A334028.
%Y A349152 - Sum and product of parts are given by A070939 and A124758.
%Y A349152 - Maximum and minimum parts are given by A333766 and A333768.
%Y A349152 Classes of standard compositions:
%Y A349152 - Partitions and strict partitions are ranked by A114994 and A333256.
%Y A349152 - Multisets and sets are ranked by A225620 and A333255.
%Y A349152 - Strict and constant compositions are ranked by A233564 and A272919.
%Y A349152 - Permutations are ranked by A333218.
%Y A349152 - Relatively prime compositions are ranked by A291166*, complement A291165.
%K A349152 nonn
%O A349152 1,3
%A A349152 _Gus Wiseman_, Nov 15 2021