# [seqfan] Re: primes beginning 101 in binary

Neil Sloane njasloane at gmail.com
Sat Sep 19 18:07:11 CEST 2015

```David, Thank you! I've edited the comment in A262350 accordingly.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Sat, Sep 19, 2015 at 11:26 AM, David Wilson <davidwwilson at comcast.net>
wrote:

> Yes, this type of sequence is infinite.
>
> By the prime number theorem, if eps > 0, there is a prime between n and
> n(1+e) for sufficient n.
> This implies that there is a prime between cb^k and (c+1)b^k for
> sufficient k.
> This means that there are an infinitude of numbers in base b starting with
> prefix c.
>
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> > Sloane
> > Sent: Saturday, September 19, 2015 10:11 AM
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] primes beginning 101 in binary
> >
> > Alois has produced a clean binary version of A089755 and A262283, namely
> > A262350:
> > a(1) = 2. For n>1, let s denote the binary string of a(n-1) with the
> leftmost 1
> > and following consecutive 0's removed. Then a(n) is the smallest prime
> not
> > yet present whose binary representation begins with s.
> >
> > This led me to look at "primes beginning 101 in binary" etc.:
> > Primes whose binary expansion begins with binary expansion of 1, 2, 3,
> 4, 5,
> > 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
> >
> > My question is, is it known that any of these sequences are infinite
> (other
> > than A000040, the primes themselves)?
> >
> > Is there a proof that the new sequence A262350 or its complement are
> > infinite?
> >
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```