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

Author

Hugo Pfoertner, Dec 17 2015

Keywords

• nonn
• more

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

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)]

Crossrefs

Cf. A003313, A253723.