cp's OEIS Frontend

A264803 Numbers with largest ratio A003313(k)/log_2(k) in the range 2^n < k < 2^(n+1).

%I A264803 #29 May 25 2020 06:29:44
%S A264803 3,7,11,29,47,71,191,379,607,1087,2103,6271,11231,18287,34303,110591,
%T A264803 196591,357887,685951,1176431,2211837,4210399,14143037,25450463,
%U A264803 46444543,89209343,155691199,298695487,550040063,1886023151
%N A264803 Numbers with largest ratio A003313(k)/log_2(k) in the range 2^n < k < 2^(n+1).
%C A264803 The corresponding addition chain lengths are given in A253723.
%C A264803 The quotient A003313(k)/log_2(k) has its conjectured maximum of 1.46347481 for k=71. Values of A003313 up to 2^31-1 are obtained from Achim Flammenkamp's web page, which provides a table computed by _Neill M. Clift_.
%C A264803 In the paper by Wattel & Jensen, the conjectured maximum is proved to hold for all k > 71, too. - _Achim Flammenkamp_, Nov 01 2016
%D A264803 E. Wattel, G. A. Jensen, Efficient calculation of powers in a semigroup, 1968 in Zuivere Wiskunde 1/68. [From _Achim Flammenkamp_, Nov 01 2016]
%H A264803 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html">Shortest addition chains</a>
%H A264803 Hugo Pfoertner, <a href="/A264803/a264803.pdf">Plot of Records of A003313(k)/log_2(k) in Intervals [2^n,2^(n+1)]</a>
%e A264803 a(3) = 11, because the maximum of quotients of shortest addition chain length l(k) and the base-2 logarithm of the numbers in the range 2^3 ... 2^4 occurs at k=11.
%e A264803   k l(k) log_2(k) l(k)/log_2(k)
%e A264803    8  3   3.0000   1.00000
%e A264803    9  4   3.1699   1.26186
%e A264803   10  4   3.3219   1.20412
%e A264803   11  5   3.4594   1.44532
%e A264803   12  4   3.5849   1.11577
%e A264803   13  5   3.7004   1.35119
%e A264803   14  5   3.8074   1.31325
%e A264803   15  5   3.9069   1.27979
%e A264803   16  4   4.0000   1.00000
%e A264803 a(30)=1886023151 because it produces the largest value of A003313(k)/log_2(k) in the interval 2^30 < k < 2^31, i.e., all other numbers in this range give a smaller quotient than A003313(1886023151) / log_2(1886023151) = 38 / 30.8127 = 1.23325771.
%Y A264803 Cf. A003313, A253723.
%K A264803 nonn,more
%O A264803 1,1
%A A264803 _Hugo Pfoertner_, Dec 17 2015