OFFSET
1,1
COMMENTS
Numbers that appear exactly 3 times in A025586, which gives the largest value in the 3x + 1 trajectory of n.
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).
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
EXAMPLE
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:
..... 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
........... 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Dec 01 2013
STATUS
approved