Problems & Puzzles: Puzzles

Puzzle 213.  Hailstone Champion Sequences

Hailstone sequences of numbers, {ai} are these produced in the so-called Collatz problem for a given starting number a1 (see 1  & 2), applying recursively the following rule:

ai+1 = 3*ai +1 if ai is odd; otherwise ai+1 = ai / 2

For each particular initial value a1 there is only one maximal member, M(a1) in the corresponding sequence.

For example if a1=7 , the hailstone sequences is:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ....

and M(7)=52

According to the rules of the Collatz problem for generating the hailstone sequences, M(a1)=a1 only if a1=2^k, for k=>2; otherwise, M(a1)>a1.

The quotient Q(a1)=M(a1)/a1 is a measure of the height of M(a1) relative to the initial value a1 of the corresponding hailstone sequence.

I have produced a table of "champion hailstone sequences", for a1 >0. In this table we annotate in the first row the triad of values {a1, M(a1), Q(a1) } for a1=1; then we annotate in the next row, the next earliest a1 value such that its Q(a1) is larger than the Q(a1) of the previous row; and so on...

 a1 M(a1) Q(a1) 1 4 4.0 3 16 5.3 7 52 7.4 15 160 10.6 27 9232 341.9 703 250504 356.3 1819 1276936 701.9 4255 6810136 1,600.5 4591 8153620 1,776.0 9663 27114424 2,806.0 26623 106358020 3,994.9 60975 593279152 9,729.8 77671 1570824736 20,224.0 ... ... ... 3873535 858555169576 221,646.4 ... ... ...

Questions:

1. Does Q(a1) grow beyond any limit?

2. Do you devise any special (non-trivial) property for the a1 values in the table of champions hailstone sequences?

Solution: William Lipp solved the question 1:

3^a*2^b-1 goes to 3^(a+1)*2^b-2 goes to 3^(a+1)*2^(b-1)-1. Hence 3^a*2^b-1 goes to 3^(a+1)*2^(b-1)-1 in two steps. Hence 2^n-1 goes, in (2n-2) steps, to 3^(n-1)*2-1and is (2n-1) steps to 3^n*2-2. Hence M(2^n-1) >= 3^n*2-1 and Q(2^n-1) >= (3^n*2-2)/(2^n-1). For large n this number is approximately 2*(3/2)^n, and it clearly increases without bound as n increases.

*** Records   |  Conjectures  |  Problems  |  Puzzles  Home | Melancholia | Problems & Puzzles | References | News | Personal Page | Puzzlers | Search | Bulletin Board | Chat | Random Link Copyright © 1999-2012 primepuzzles.net. All rights reserved.