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

40, 64, 100, 112, 136, 148, 184, 208, 244, 256, 280, 352, 400, 424, 472, 532, 544, 616, 640, 688, 712, 724, 784, 820, 832, 868, 904, 928, 964, 976, 1048, 1072, 1108, 1120, 1156, 1192, 1216, 1264, 1300, 1360, 1396, 1408, 1432, 1480, 1540, 1576, 1588, 1624, 1684

Offset: 1

Author

Jon E. Schoenfield, Dec 01 2013

Keywords

• nonn

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).

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.

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.

Examples

			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
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A025586, A213199, A222562, A233293.