OFFSET
1,1
COMMENTS
Also numbers k such that the excess compression of the k-th integer composition in standard order is 1.
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.
postn of 1 in
LINKS
EXAMPLE
The terms and corresponding compositions begin:
3: (1,1)
11: (2,1,1)
14: (1,1,2)
19: (3,1,1)
27: (1,2,1,1)
28: (1,1,3)
29: (1,1,2,1)
35: (4,1,1)
46: (2,1,1,2)
51: (1,3,1,1)
56: (1,1,4)
57: (1,1,3,1)
67: (5,1,1)
75: (3,2,1,1)
78: (3,1,1,2)
83: (2,3,1,1)
91: (2,1,2,1,1)
92: (2,1,1,3)
93: (2,1,1,2,1)
99: (1,4,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[100], Total[stc[#]] == Total[First/@Split[stc[#]]]+1&]
CROSSREFS
These compositions are counted by A373950.
Positions of ones in A373954.
A003242 counts compressed compositions (or anti-runs).
A114901 counts compositions with no isolated parts.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 encodes compression using compositions in standard order.
A373949 counts compositions by compression-sum.
A373953 gives compression-sum of standard compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 29 2024
STATUS
approved