cp's OEIS Frontend

A232870 Numbers that are the largest value in the Collatz (3x+1) trajectories of exactly three initial values.

%I A232870 #17 Dec 07 2013 16:35:57
%S A232870 40,64,100,112,136,148,184,208,244,256,280,352,400,424,472,532,544,
%T A232870 616,640,688,712,724,784,820,832,868,904,928,964,976,1048,1072,1108,
%U A232870 1120,1156,1192,1216,1264,1300,1360,1396,1408,1432,1480,1540,1576,1588,1624,1684
%N A232870 Numbers that are the largest value in the Collatz (3x+1) trajectories of exactly three initial values.
%C A232870 Numbers that appear exactly 3 times in A025586, which gives the largest value in the 3x + 1 trajectory of n.
%C A232870 For each term k in this sequence, the three initial values, that is, values of n at which A025586(n) = k, are (in ascending order) n1 = (k-1)/3, n2 = 2*n1 = 2*(k-1)/3, and n3 = k. n1 is the odd number from which an upward (that is, 3x + 1) step lands at k = 3*n1 + 1. It cannot be the case that n1 = 3 (mod 4), because we would then have k = 10 (mod 12), so k/2 would be odd, and its successor in the trajectory would be 3*k/2 + 1 > k, so k would not be the largest value in the trajectory. Thus, n1 = 1 (mod 4), so n2 = 2 (mod 8) and n3 = 4 (mod 12).
%C A232870 Numbers that are the largest value in the 3x + 1 trajectory of exactly one initial value (that is, numbers that appear exactly once in A025586) are in A222562.
%C A232870 Numbers that are not the largest value in the 3x + 1 trajectory of any initial value (that is, numbers that do not appear at all in A025586) are in A213199.
%H A232870 Jon E. Schoenfield, <a href="/A232870/b232870.txt">Table of n, a(n) for n = 1..10000</a>
%e A232870 40 is in the sequence because it is the largest value in the 3x + 1 trajectories of exactly three initial values: 13, 26, and 40 itself. The trajectories are as follows:
%e A232870 ..... 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
%e A232870 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
%e A232870 ........... 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
%Y A232870 Cf. A025586, A213199, A222562, A233293.
%K A232870 nonn
%O A232870 1,1
%A A232870 _Jon E. Schoenfield_, Dec 01 2013