A365482 In the Collatz (3x+1) problem, values in A006884 for which the maximum excursion ratio (see comments) is greater than 2.

27, 319804831, 1410123943, 3716509988199, 9016346070511, 1254251874774375, 10709980568908647, 1980976057694848447

Offset

1,1

Comments

Kontorovich and Lagarias (2009, 2010) define the maximum excursion ratio as the ratio between the log of the highest point in the trajectory of the T function (started at x) and the log of x, where T(x) is the 3x+1 function = (3x+1)/2 if x is odd, x/2 if x is even (A014682).

They use data from Oliveira e Silva (2010) to compile Table 3 in their paper, but they omit the a(7) = 10709980568908647 value (cf. also Barina and Roosendall links).

Equivalently, values in A006884 for which A365478(A006884(k)) / A006884(k)^2 > 1, for k >= 1.

See A365483 for corresponding maximum excursion values.

Links

Table of n, a(n) for n=1..8.

David Barina, Path records.

Alex V. Kontorovich and Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems, arXiv:0910.1944 [math.NT], 2009, pp. 11-14, and in Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 140-142.

Tomás Oliveira e Silva, Empirical Verification of the 3x+1 and Related Conjectures, in Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 189-207.

Eric Roosendall, 3x + 1 Path Records.

Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Subsequence of A006884.

Cf. A014682, A365478, A365483.

Sequence in context: A106536 A227115 A295189 * A243105 A023925 A022067

Adjacent sequences: A365479 A365480 A365481 * A365483 A365484 A365485

Keyword

nonn,hard,more

Author

Paolo Xausa, Sep 05 2023

Status

approved