

A264803


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


2



3, 7, 11, 29, 47, 71, 191, 379, 607, 1087, 2103, 6271, 11231, 18287, 34303, 110591, 196591, 357887, 685951, 1176431, 2211837, 4210399, 14143037, 25450463, 46444543, 89209343, 155691199, 298695487, 550040063, 1886023151
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OFFSET

1,1


COMMENTS

The corresponding addition chain lengths are given in A253723.
The quotient A003313(k)/log_2(k) has its conjectured maximum of 1.46347481 for k=71. Values of A003313 up to 2^311 are obtained from Achim Flammenkamp's web page, which provides a table computed by Neill M. Clift.
In the paper by Wattel & Jensen, the conjectured maximum is proved to hold for all k > 71, too.  Achim Flammenkamp, Nov 01 2016


REFERENCES

E. Wattel, G. A. Jensen, Efficient calculation of powers in a semigroup, 1968 in Zuivere Wiskunde 1/68. [From Achim Flammenkamp, Nov 01 2016]


LINKS



EXAMPLE

a(3) = 11, because the maximum of quotients of shortest addition chain length l(k) and the base2 logarithm of the numbers in the range 2^3 ... 2^4 occurs at k=11.
k l(k) log_2(k) l(k)/log_2(k)
8 3 3.0000 1.00000
9 4 3.1699 1.26186
10 4 3.3219 1.20412
11 5 3.4594 1.44532
12 4 3.5849 1.11577
13 5 3.7004 1.35119
14 5 3.8074 1.31325
15 5 3.9069 1.27979
16 4 4.0000 1.00000
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.


CROSSREFS



KEYWORD

nonn,more


AUTHOR



STATUS

approved



