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A339614 Inputs n that yield a record-breaking value of A008908(n)/(log_2(n)+1) for the Collatz conjecture.

%I A339614 #37 Aug 21 2021 05:54:19
%S A339614 1,3,7,9,27,26623,35655,52527,142587,156159,230631,626331,837799,
%T A339614 1723519,3542887,3732423,5649499,6649279,8400511,63728127,
%U A339614 3743559068799,100759293214567,104899295810901231
%N A339614 Inputs n that yield a record-breaking value of A008908(n)/(log_2(n)+1) for the Collatz conjecture.
%C A339614 The metric A008908(n)/(log_2(n)+1) is always equal to 1 for any power of 2 (where 1 is the smallest possible value).
%H A339614 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A339614 a(1) = 1, which is trivial, because the first element in any sequence is record setting.
%e A339614 a(5) = 27, because A008908(n)/(log_2(n)+1) yields a maximum value at n=27 among the first 27 elements, and there are 4 record-breaking elements beforehand.
%o A339614 (Python)
%o A339614 import math
%o A339614 oeis_A006877 = [1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 10623, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, 1117065, 1501353, 1723519, 2298025, 3064033, 3542887, 3732423, 5649499, 6649279, 8400511, 11200681, 14934241, 15733191, 31466382, 36791535, 63728127]
%o A339614 def stopping_time(n):
%o A339614     time = 1
%o A339614     while n>1:
%o A339614         n = 3*n + 1 if n & 1 else n//2
%o A339614         time += 1
%o A339614     return time
%o A339614 def stopping_time_metric(n):
%o A339614     time = stopping_time(n)
%o A339614     logarithmic_distance = (math.log(n, 2)+1)
%o A339614     return float(time/logarithmic_distance)
%o A339614 result = []
%o A339614 record_input = oeis_A006877[0]
%o A339614 record_stopping_time_metric = stopping_time_metric(record_input)
%o A339614 result.append(record_input)
%o A339614 for n in range(1, len(oeis_A006877)):
%o A339614     current_input = oeis_A006877[n]
%o A339614     current_stopping_time_metric = stopping_time_metric(current_input)
%o A339614     if current_stopping_time_metric > record_stopping_time_metric:
%o A339614         record_input = current_input
%o A339614         record_stopping_time_metric = current_stopping_time_metric
%o A339614         result.append(record_input)
%o A339614 for n in range(len(result)):
%o A339614     print(result[n], end=", ")
%Y A339614 Cf. A008908, A006877, A033492.
%K A339614 nonn,more
%O A339614 1,2
%A A339614 _Matthew Russell Downey_, Dec 10 2020