@Happy5214 : Yes, I had noticed for the presence of 797161 but I hadn't seen that 797161 = (2*398581-1) !

I also just noticed that 3^13 ends directly on the prime number 797161 at index 1.

Thank you for checking the conjecture even further !

@EdH : Thank you for these precious tables !

I'm going to look at them very closely tomorrow.

But in just a few minutes I have already noticed exactly the same phenomenon with the 37-digit prime number : 1535090713229126909942383374434289901 which is in the decomposition of all terms in index 1 and 2 of all the sequences that start with 3^(206*k), k integer.

And exactly in the same way, 3^103 also ends directly on a prime number (of 49 digits) : 69575965298821529689922251835887181478451547013. On the other hand, I haven't yet found the relation between these two prime numbers, as Happy5214 did for the previous case !

It's this line of the table that made me see this new similar case :

Code:

prime 1535090713229126909942383374434289901 shows up 2 times (206:i1, 206:i2).

I just checked this new conjecture up to 3^824 and it works fine !