A324248 Odd numbers with dropping time of the reduced Collatz iteration (A122458) exceeding 5.

27, 31, 47, 63, 71, 91, 103, 111, 127, 155, 159, 167, 191, 207, 223, 231, 239, 251, 255, 283, 287, 303, 319, 327, 347, 359, 367, 383, 411, 415, 423, 447, 463, 479, 487, 495, 507, 511, 539, 543, 559, 575, 583, 603, 615, 623, 639, 667, 671, 679, 703, 719, 735, 743, 751, 763, 767

Offset: 1

Author

Wolfdieter Lang, Feb 21 2019

Keywords

• nonn
• easy

Comments

Note that the Collatz conjecture is assumed. Otherwise there may exist (very large) odd numbers for which no finite dropping time exists

This sequence is obtained from the residue classes modulo 256 of the entries a(1) to a(19): 27, 31, 47, 63, 71, 91, 103, 111, 127, 155, 159, 167, 191, 207, 223, 231, 239, 251, 255. See the Klee-Wagon reference where the function C is C(2*n+1) = A075677(n+1) = A000265(3*n+2), for n >= 0, and dropping time is called there stopping time (on p. 225 Exercise 4 should be Exercise 5 given on p. 229, with hints on p. 309; also in the first line after (1) 'less than 5' should be replaced by 'not exceeding 5').

The values of (a(j)-1)/2, for j = 1..19, are 13, 15, 23, 31, 35, 45, 51, 55, 63, 77, 79, 83, 95, 103, 111, 115, 119, 125, 127.

The dropping times are 5 for the seven residue classes modulo 256 of 39, 79, 95, 123, 175, 199, and 219. See Klee-Wagon, Exercise 5 (a), p. 229.

The dropping times for a(n) are given in A324249(n).

Examples

			n = 1: The trajectory under the reduced Collatz function fr for (a(1) - 1)/2  = 13 is given as an example in A122456, from which the dropping time is read off as 37 = A122456(13).
n = 2: The dropping time of a(2) = 31 is 35 = A122456(15). The second to last trajectory number 5 is the first number < 31.
		

References

• Victor Klee and Stan Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America (1991) pp. 191-194, 225-229, 308-309.

Crossrefs

Cf. A000265, A075677, A122458, A324249.

Formula

Sorted sequence of the odd numbers 2*m + 1 with least positive integer k >= 6 such that fr^[k](m) < 2*m + 1, for m >= 1, where fr is the reduced Collatz function fr(m) := A075677(m+1) = A000265(3*m+2).