[seqfan] Re: Jumping off the primes thread

[israel at math.ubc.ca]

The n'th prime p(n) has about log_{10}(p(n)) decimal digits.
The probability of its matching a random number of this size is 
about 1/10^log_{10}(p(n)) = 1/p(n). Since sum_n 1/p(n) diverges,
we should expect it to match infinitely many times.  However, 
since that sum diverges extremely slowly, these matches should
be very sparse.

Cheers,
Robert

On Mar 22 2020, Daniel via SeqFan wrote:

> The title of the current conversation about "primes describing digit 
> positions" gave me an idea: When does the nth prime number appear 
> starting with the nth digit of pi? A good starting point is to look for 
> cases where A064467(n)=n, but there are none in that sequence's b-file. 
> However, a given prime might appear both before and at the nth digit of 
> pi, so I compared the primes and the digits of pi directly.
>
> My Excel spreadsheet couldn't find any examples up to n=10,000. But I did 
> find a few near-misses: * Obviously the 2nd prime number (3) appears in 
> the 1st digit of pi. * The 41st prime number is 179 - and digits 41-43 of 
> pi are 169. Just one off on the middle digit! * The 63rd prime number 
> (307) misses by only two places: it appears in digits 65-67 of pi. * The 
> 118th prime number (647) misses by only one place: it appears in digits 
> 119-121 of pi.
>
> Obviously the probability of finding one shrinks as the numbers climb. 
> Are there any examples at all? If so, are they sequence-worthy? If not, 
> is there an underlying reason why not? Daniel Sterman
>
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