|
%I #29 May 25 2020 06:29:44
%S 3,7,11,29,47,71,191,379,607,1087,2103,6271,11231,18287,34303,110591,
%T 196591,357887,685951,1176431,2211837,4210399,14143037,25450463,
%U 46444543,89209343,155691199,298695487,550040063,1886023151
%N Numbers with largest ratio A003313(k)/log_2(k) in the range 2^n < k < 2^(n+1).
%C The corresponding addition chain lengths are given in A253723.
%C 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 In the paper by Wattel & Jensen, the conjectured maximum is proved to hold for all k > 71, too. - _Achim Flammenkamp_, Nov 01 2016
%D 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 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html">Shortest addition chains</a>
%H 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 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 k l(k) log_2(k) l(k)/log_2(k)
%e 8 3 3.0000 1.00000
%e 9 4 3.1699 1.26186
%e 10 4 3.3219 1.20412
%e 11 5 3.4594 1.44532
%e 12 4 3.5849 1.11577
%e 13 5 3.7004 1.35119
%e 14 5 3.8074 1.31325
%e 15 5 3.9069 1.27979
%e 16 4 4.0000 1.00000
%e 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 Cf. A003313, A253723.
%K nonn,more
%O 1,1
%A _Hugo Pfoertner_, Dec 17 2015
|
