TITLE: Collatz conjecture for numbers of th form $2^n +1$
QUESTION [28 upvotes]: 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 $2^n-1$

REPLY [7 votes]: 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.