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A140571
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Decimal expansion of the universal constant in E(h), the maximum number of essential elements of order h.
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0
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2, 0, 5, 7, 2, 8, 4, 1, 2, 8, 4, 7, 8, 7, 9, 3, 4, 1, 2, 8, 5, 8, 2, 2, 3, 9, 6, 4, 4, 8, 3, 7, 6, 9, 0, 9, 1, 0, 0, 4, 3, 4, 7, 8, 2, 7, 4, 9, 4, 2, 1, 2, 6, 8, 0, 7, 4, 1, 5, 3, 8, 1, 9, 6, 6, 2, 4, 2, 3, 6, 9, 2, 9, 5, 4, 2, 7, 6, 3, 5, 1, 3, 3, 4, 9, 8, 5, 1, 9, 0, 8, 0, 7, 8, 9, 0, 1, 6, 5, 3, 6, 5, 5, 9, 7, 7
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OFFSET
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1,1
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COMMENTS
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A fundamental result of Erdos and Graham is that every integer basis possesses only finitely many essential elements. Grekos refined this, showing that the number of essential elements in a basis or order h is bounded by a function of h only. Deschamps and Farhi (2007) proved a best possible upper bound on this function, which contains a constant whose digits are this sequence.
Abstract: Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h,k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that
E(h,k) = Theta_{k} ([h^{k}/log h]^{1/(k+1)}). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known "postage stamp problem" in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show that E(h,k) ~ (h-1) (log k)/(log log k).
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LINKS
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FORMULA
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Equals 30*sqrt(log(1564)/1564).
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EXAMPLE
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2.0572841284787934...
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MATHEMATICA
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RealDigits[(30*Sqrt[Log[1564]/1564]), 10, 120][[1]] (* Harvey P. Dale, Sep 27 2023 *)
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PROG
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CROSSREFS
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Cf. postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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