# [seqfan] Re: Jumping off the primes thread

Neil Fernandez primeness at borve.org
Sun Mar 22 21:38:28 CET 2020

```Hi Daniel,

I found no examples up to n=2*10^5.

Mathematica
For[x = 1, x < 200001, x++,
t = Prime[x];
k = IntegerLength[t];
q = FromDigits[Row[First at RealDigits[Pi, 10, k, 1 - x]][[1]]];
If[t == q, Print[x], ""]]

Cases of the nth prime running from the nth digit of y:

y=pi] no solutions < 2*10^5;

y=e] n=1; no others < 10^5;

y=sqrt(2)] n=6; no others < 10^5;

y=sqrt(3)] no solutions < 10^5;

y=sqrt(5)] n=1, n=31, n=467; no others < 10^5;

y=sqrt(7)] n=1, n=17, n=3094; no others < 10^5.

We might want to look at the sequence

a(n):= smallest x such that the xth prime runs from the xth digit of the
square root of the nth prime, or 0 if no x is known.

This sequence begins

6,0,1,1,...

We can also define

a(n):= largest known x such that the xth prime runs from the xth digit
of the square root of the nth prime, or 0 if no x is known.

This begins

6,0,467,3094,...

Neil

In message <1395125322.73434.1584862878192 at mail.yahoo.com>, Daniel via
SeqFan <seqfan at list.seqfan.eu> writes
>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

--
Neil Fernandez

```