A264803 Numbers with largest ratio A003313(k)/log_2(k) in the range 2^n < k < 2^(n+1).
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
Offset: 1
Examples
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.
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
- Achim Flammenkamp, Shortest addition chains
- Hugo Pfoertner, Plot of Records of A003313(k)/log_2(k) in Intervals [2^n,2^(n+1)]
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