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 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 (list; graph; refs; listen; history; text; internal format)
 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 2^31-1 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 Table of n, a(n) for n=1..30. Achim Flammenkamp, Shortest addition chains Hugo Pfoertner, Plot of Records of A003313(k)/log_2(k) in Intervals [2^n,2^(n+1)] EXAMPLE 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. 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 Cf. A003313, A253723. Sequence in context: A194373 A156210 A343718 * A117790 A014447 A233517 Adjacent sequences: A264800 A264801 A264802 * A264804 A264805 A264806 KEYWORD nonn,more AUTHOR Hugo Pfoertner, Dec 17 2015 STATUS approved

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Last modified June 13 03:56 EDT 2024. Contains 373366 sequences. (Running on oeis4.)