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A046693
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Size of smallest subset S of N={0,1,2,...,n} such that S-S=N, where S-S={abs(i-j) | i,j in S}.
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10
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1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16
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OFFSET
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0,2
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COMMENTS
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It is easy to show that a(n+1) must be no larger than a(n)+1. Problem: Can a(n+1) ever be smaller than a(n)?
Problem above solved in A103300. a(137) smaller than a(136).
Except for initial term, round(sqrt(3*n + 9/4)) up to n=51. See A308766 for divergences up to n=213. See A326499 for a list of best known solutions.
From Ed Pegg Jr, Jun 23 2019: (Start)
Minimal marks for a sparse ruler of length n.
Minimal vertices in a graceful graph with n edges. (End)
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LINKS
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EXAMPLE
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a(10)=6 since all integers in {0,1,2...10} are differences of elements of {0,1,2,3,6,10}, but not of any 5-element set.
a(17)=7 since all integers in {0,1,2...17} are differences of elements of {0,1,8,11,13,15,17}, but not of any 6-element set.
In other words, {0,1,8,11,13,15,17} is a restricted difference basis w.r.t. A004137(7)=17.
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MATHEMATICA
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Prepend[Table[Round[Sqrt[3*n+9/4]]+If[MemberQ[A308766, n], 1, 0], {n, 1, 213}], 1]
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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STATUS
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approved
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