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Let's define a prime number to be considered '$Special $ ' if the number of steps it takes to reach $1$ in the collatz sequence is a perfect square of a prime number.

For example , $13$ could be called ' $Special $ ' since it takes $ 9 $ steps to reach $1$.

When observing such numbers , I found the trend that for higher numbers , If a number is ' $Special $ ' , then it is very likely that the ' $Special $ ' numbers following it also take same number of steps .

For example , between $1258837$ and $1551577$ , we have $166$ consecutive ' $Special $ ' numbers which takes $49$ steps to reach to 1.

Why is there regularity between number of steps of such number ? Also is there a prime number whose square is not a number of step of such numbers ?

P.S: Can anyone suggest how can I show that data on the website ? (the data is in the python IDLE output and is quite big.)

The Demonix _ Hermit
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  • Likely to be helpful: https://math.stackexchange.com/questions/470782/longest-known-sequence-of-identical-consecutive-collatz-sequence-lengths – Klangen Oct 07 '19 at 08:39
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    @Klangen I have seen that question and I can assure that the question I've asked is completely different than that. – The Demonix _ Hermit Oct 07 '19 at 08:40
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    To me it sounds very similar, as there are also many consecutive non-''Special'' numbers with identical stopping times. It is a known phenomenon. – Klangen Oct 07 '19 at 09:01
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    @TheDemonix_Hermit Regarding the Python data, perhaps you could link to another webpage with the data ? The phenomenon of "primes with squared primes stopping time" seems far-fetched and probably rare and uninteresting, so I guess you need to back up your idea some statistical confirmation in order to make other people interested in it. In the range you looked at, what percentage of the"usual", well-known phenomenon (identical stopping times for succesive numbers) correspond to your phenomenon ? – Ewan Delanoy Oct 20 '19 at 15:37
  • Here is a reference to the corresponding OEIS sequence. – jorisperrenet Jul 12 '23 at 18:13

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