OFFSET
1,1
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).
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.
FORMULA
Equals 30*sqrt(log(1564)/1564).
EXAMPLE
2.0572841284787934...
MATHEMATICA
RealDigits[(30*Sqrt[Log[1564]/1564]), 10, 120][[1]] (* Harvey P. Dale, Sep 27 2023 *)
PROG
(PARI) 30*sqrt(log(1564)/1564) \\ Michel Marcus, Oct 18 2018
CROSSREFS
KEYWORD
AUTHOR
Jonathan Vos Post, Jul 05 2008
EXTENSIONS
a(100) corrected by Georg Fischer, Jul 12 2021
STATUS
approved