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A354904
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Numbers k such that the k-th composition in standard order is not the sequence of run-sums of any other composition.
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6
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3, 7, 11, 14, 15, 19, 23, 27, 28, 29, 30, 31, 35, 39, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 75, 78, 79, 83, 87, 91, 92, 93, 94, 95, 99, 103, 107, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127
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OFFSET
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1,1
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COMMENTS
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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.
Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).
The first term k such that the k-th composition in standard order does not have ones sandwiching the same prime number an even number of times is k = 3221, corresponding to the composition (1,3,3,2,2,1).
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LINKS
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EXAMPLE
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The terms and their corresponding compositions begin:
3: (1,1)
7: (1,1,1)
11: (2,1,1)
14: (1,1,2)
15: (1,1,1,1)
19: (3,1,1)
23: (2,1,1,1)
27: (1,2,1,1)
28: (1,1,3)
29: (1,1,2,1)
30: (1,1,1,2)
31: (1,1,1,1,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], FreeQ[Total/@Split[#]&/@ Join@@Permutations/@IntegerPartitions[Total[stc[#]]], stc[#]]&]
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CROSSREFS
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These compositions are counted by A354909.
A124767 counts runs in standard compositions.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353852 ranks compositions with distinct run-sums, counted by A353850.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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