cp's OEIS Frontend

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

%I A140571 #18 Sep 27 2023 12:21:05
%S A140571 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,
%T A140571 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,
%U A140571 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
%N A140571 Decimal expansion of the universal constant in E(h), the maximum number of essential elements of order h.
%C A140571 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.
%C A140571 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
%C A140571 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).
%H A140571 Bruno Deschamps, Bakir Farhi, <a href="https://doi.org/10.1016/j.jnt.2006.06.002">Essentialité dans les bases additives</a>, J. Number Theory, 123 (2007), p. 170-192.
%H A140571 P. Erdos and R. L. Graham, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa37/aa37119.pdf">On Bases with an Exact Order</a>, Acta Arith. 37(1980)201-207.
%H A140571 G. Grekos, <a href="https://www.jstor.org/stable/44166477">Sur l'ordre d'une base additive</a>, (French) Séminaire de Théorie des nombres de Bordeaux, 1987/1988, exposé 31.
%H A140571 Peter Hegarty, <a href="http://arxiv.org/abs/0807.0463">The Postage Stamp Problem and Essential Subsets in Integer Bases</a>, arXiv:0807.0463 [math.NT], 2008.
%F A140571 Equals 30*sqrt(log(1564)/1564).
%e A140571 2.0572841284787934...
%t A140571 RealDigits[(30*Sqrt[Log[1564]/1564]),10,120][[1]] (* _Harvey P. Dale_, Sep 27 2023 *)
%o A140571 (PARI) 30*sqrt(log(1564)/1564) \\ _Michel Marcus_, Oct 18 2018
%Y A140571 Cf. postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.
%K A140571 cons,easy,nonn
%O A140571 1,1
%A A140571 _Jonathan Vos Post_, Jul 05 2008
%E A140571 a(100) corrected by _Georg Fischer_, Jul 12 2021