[seqfan] Jumping off the primes thread

Daniel kimpire at yahoo.com
Sun Mar 22 08:41:18 CET 2020

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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