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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