A140571 Decimal expansion of the universal constant in E(h), the maximum number of essential elements of order h.

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

Offset: 1

Author

Jonathan Vos Post, Jul 05 2008

Keywords

• cons
• easy
• nonn

Comments

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).

Examples

			2.0572841284787934...
		

Links

• Bruno Deschamps, Bakir Farhi, Essentialité dans les bases additives, J. Number Theory, 123 (2007), p. 170-192.
• P. Erdos and R. L. Graham, On Bases with an Exact Order, Acta Arith. 37(1980)201-207.
• G. Grekos, Sur l'ordre d'une base additive, (French) Séminaire de Théorie des nombres de Bordeaux, 1987/1988, exposé 31.
• Peter Hegarty, The Postage Stamp Problem and Essential Subsets in Integer Bases, arXiv:0807.0463 [math.NT], 2008.

Crossrefs

Cf. postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Programs

Mathematica

RealDigits[(30*Sqrt[Log[1564]/1564]),10,120][[1]] (* Harvey P. Dale, Sep 27 2023 *)

PARI

30*sqrt(log(1564)/1564) \\ Michel Marcus, Oct 18 2018

Formula

Equals 30*sqrt(log(1564)/1564).

Extensions

a(100) corrected by Georg Fischer, Jul 12 2021