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Everybody has heard of the Collatz conjecture and it is a nice programming exercise to write a function, that calculates for a given number $n$ the number of iterations it takes until one reaches $1$. However if one restricts to numbers of the form $2^n+1$ one gets the following sequence of integers (NO matches in oeis). It starts with

7,5,19,12,26,27,121,122,35,36,156,113,52,53,98,99,100,101,102, 72,166,167,168,169,170,171,247,173,187,188,251,252,178,179,317, 243,195,196,153,154,155,156,400,326,495,496,161,162,331,332,408, 471,410,411,337,338,339,340,553,479,480,481,482,483,559,560,561, 562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,626, 578,628,629,630,631,583,584,634,635,636,637,894,895,640,641,898,643

"Usually" it grows by $1$ and at some positions it takes a completely different value. Then sometimes it jumps back as if there was never a different value involved (like 575,576,626,578) This seemed to me a bit strange/interesting and funny.It there anything known about this special sequence. Maybe there is a characterization of those positions, where this sequence grows by $1$. I am not sure, how to make a well posed question out of this.

EDIT: and there is the same behavior for numbers of the form $2^n-1$

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I'm sorry -- I feel a little stupid -- but I am not understanding this. Let me ask a dumb question: why does this start with 7, and why is 5 the next number? –  Todd Trimble Jan 4 '11 at 15:55
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Because starting with 3 takes 7 steps to get to 1, and starting with 5 takes 5 steps. –  Thomas Bloom Jan 4 '11 at 15:58
    
After $4 k= 4*\lfloor \frac{n}{2}\rfloor$ steps, you end up with either $3^k+1$ or $2*3^k +1$, with the former if $n$ is even and the latter if $n$ is odd. There seem to be some interesting relations between Collatz sequences of these numbers, but I'm not an expert at all, so I know nothing of research done in this field. Same holds for $2^n -1$, ends up at $3^k-1$ or $2*3^k -1$ after same $k$ steps. –  Jan Jitse Venselaar Jan 4 '11 at 16:23
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Thank you, Thomas. –  Todd Trimble Jan 4 '11 at 16:37
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Henrik, your question has produced a new sequence on oeis :-) oeis.org/A179118 –  Martin Brandenburg Jun 8 '11 at 6:51

1 Answer 1

You can find some closed formulas resembling your problem in

Andrei, S., Kudlek, M., Niculescu, R., S.: Some results on the Collatz problem. Acta Informatica No. 37, Vol. 2, pp. 145-160 (2000)

http://www.springerlink.com/content/aw10yk5gr59l0n1e/

Hope it helps.

EDIT: (To complement the answer) You should also check the references in the chapter I.8 ``Consecutive numbers with the same height" in

The Dynamical System Generated by the 3n +1 Function - Gunther J. Wirsching - Lec. Notes in Mathematics v. 1681

There you'll find that there is a infinite number of very long sequences of consecutive numbers for which your Collatz function is constant, which is also very interesting.

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