Fascinating. Contrary to what I said at Re: Re (tilly) 1: Rotationally Prime Numbers, when I consider numbers that are strings of 1's, the prime number theorem naively suggests that there should be an infinite number of rotational primes of that form. (This differs from most numbers in that there is only 1 number in the rotation - the odds of 1 number being prime is far different than the odds of n of them being prime.)
Of course there are a lot of special factorization properties of long strings of 1's. So it may not be so simple as all that. But the number of rotational factors between length 23 and 1031 matches the naive prediction surprisingly well. (The naive prediction is that the number of primes out to length n should be roughly log(n)/log(10), and the number in any interval should be the difference of those two. From 23 to 1031 we'd predict 1.65 and actually had 2.)
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