cp's OEIS Frontend

A248037 Numbers n such that the ratio of tripling steps to halving steps in the Collatz (3x+1) trajectory of n is greater than all previous ratios.

%I A248037 #100 May 20 2018 03:13:12
%S A248037 2,3,7,9,27,230631,626331,837799,1723519,3732423,5649499,6649279,
%T A248037 8400511,63728127,3743559068799,100759293214567,104899295810901231
%N A248037 Numbers n such that the ratio of tripling steps to halving steps in the Collatz (3x+1) trajectory of n is greater than all previous ratios.
%C A248037 Equivalently, numbers n such that A006667(n)/A064433(n) > A006667(m)/A064433(m) for all 0 < m < n.
%C A248037 A006667(n) is the number of tripling steps in the Collatz (3x+1) problem and A064433(n) is the number of halving steps in the Collatz (3x+1) problem.
%C A248037 It is crucial to make A006667(n) the numerator as it can be zero when n = 2^k for some k > 0.
%C A248037 a(n) is odd for all n > 1.
%C A248037 The corresponding ratios are:
%C A248037 0.0000000000000000000000000000... (2)
%C A248037 0.4000000000000000000000000000... (3)
%C A248037 0.4545454545454545454545454545... (7)
%C A248037 0.4615384615384615384615384615... (9)
%C A248037 0.5857142857142857142857142857... (27)
%C A248037 0.5899280575539568345323741007... (230631)
%C A248037 0.5924764890282131661442006269... (626331)
%C A248037 0.5927051671732522796352583586... (837799)
%C A248037 0.5931232091690544412607449856... (1723519)
%C A248037 0.5935828877005347593582887700... (3732423)
%C A248037 0.5937500000000000000000000000... (5649499)
%C A248037 0.5961538461538461538461538461... (6649279)
%C A248037 0.5967365967365967365967365967... (8400511)
%C A248037 0.6030405405405405405405405405... (63728127)
%C A248037 0.6035196687370600414078674948... (3743559068799)
%C A248037 If we define a "tripling step" to also include a "halving step" afterwards (since 3*n+1 converts an odd number n into an even number, so a halving step will always follow), the ratios are still maximum at the a(n) values. However, the ratios themselves are different. The corresponding ratios in this case are:
%C A248037 0.000000000000000000000000000... (2)
%C A248037 0.666666666666666666666666666... (3)
%C A248037 0.833333333333333333333333333... (7)
%C A248037 0.857142857142857142857142857... (9)
%C A248037 1.413793103448275862068965517... (27)
%C A248037 1.438596491228070175438596491... (230631)
%C A248037 1.453846153846153846153846153... (626331)
%C A248037 1.455223880597014925373134328... (837799)
%C A248037 1.457746478873239436619718309... (1723519)
%C A248037 1.460526315789473684210526315... (3732423)
%C A248037 1.461538461538461538461538461... (5649499)
%C A248037 1.476190476190476190476190476... (6649279)
%C A248037 1.479768786127167630057803468... (8400511)
%C A248037 1.519148936170212765957446808... (63728127)
%C A248037 1.656946826758147512864493997... (3743559068799)
%C A248037 From _Jon E. Schoenfield_, Nov 21 2015: (Start)
%C A248037 Let T and H be the number of tripling steps and halving steps, respectively, in the entire Collatz (3x+1) trajectory of a number n. Since each tripling step increases the value by a factor greater than 3, and each halving step decreases it by a factor of exactly 2, we have n * 3^T / 2^H < 1, from which T/H < log(2)/log(3) - log_3(n)/H, so the ratio T/H cannot exceed log(2)/log(3) = 0.6309297535...
%C A248037 It seems likely that the present sequence is a subsequence of A006877 (which consists of values n whose trajectories are of record length). Taking as values of n the terms from the b-file for A006877, and generating their trajectories to obtain the corresponding values of H(n), it does not seem clear whether log_3(n)/H(n) is converging toward zero or toward some positive limit, perhaps around 0.020 (which would mean T/H < log(2)/log(3) - 0.020, i.e., T/H < 0.611).
%C A248037 The known terms n in A006877 at which log_3(n)/H(n) reaches a record low coincide almost exactly with the known terms of this sequence, the only exception thus far being a(11) = A006877(52) = 5649499, at which log_3(n)/H(n) is only slightly larger than it is at a(10) = A006877(51) = 3732423 (0.03685302 vs. 0.03682956). Given the values of log_3(n)/H(n) for the remaining known terms in A006877, it seems likely that
%C A248037      a(16) = A006877(110) = 100759293214567
%C A248037   and that a(17) exceeds A006877(130), which is 46785696846401151.
%C A248037 (End)
%C A248037 Note that a(17)=104899295810901231 has now been found by Eric Roosendaal's distributed project (see link below). - _Dmitry Kamenetsky_, Sep 23 2016
%C A248037 For n>=14, a(n) must be 7, 15, 27, or 31 (mod 32). This is because all other values provably have a ratio of tripling to halving steps of less than 0.6 (see program by Irvine and Consiglio Jr.). - _Dmitry Kamenetsky_, Sep 24 2016
%H A248037 Sean A. Irvine and David Consiglio, Jr., <a href="/A248037/a248037.c.txt">C program</a>
%H A248037 Eric Roosendaal, <a href="http://www.ericr.nl/wondrous/comprecs.html">Completeness and Gamma records</a>
%H A248037 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%o A248037 (PARI)
%o A248037 Tratio(n)=c=0; d=0; while(n!=1,if(n==Mod(0,2),n=n/2;d++); if(n==Mod(1,2)&&n!=1,n=3*n+1;c++));c/d
%o A248037 print1(2,", "); n=2; p=Tratio(2); while(n,t=Tratio(n+1); if(p>=t,n+=2); if(p<t,print1(n+1,", "); p=t; n+=2))
%Y A248037 Cf. A006667, A006877, A008908, A064433.
%K A248037 nonn,more,hard
%O A248037 1,1
%A A248037 _Derek Orr_, Sep 29 2014
%E A248037 Corrected and extended by _Sean A. Irvine_, _Derek Orr_, and _David Consiglio, Jr._, Nov 23 2015
%E A248037 a(16) from _David Consiglio, Jr._ and _Sean A. Irvine_, Nov 26 2015
%E A248037 a(17) added by _Dmitry Kamenetsky_, Sep 23 2016