[seqfan] Re: Smallest prime with a substring of exactly n zeros

[Bob Selcoe]

Robert, that is very interesting.

I would now conjecture that the number of terms without all contiguous zeros 
exceed those with them.  Undoubtedly a proof would be quite challenging.

It would also be interesting to see if any terms are have three or more 
substrings of zeros.  My guess is yes (with first appearance at n much 
larger than 900) for essentially the same reason as the initial question 
about all contiguous zeros, but I don't have any real insight into the 
matter beyond that.  In lieu of identifying any such term(s), does anyone 
have any ideas about this?

Cheers,
Bob S
PS - just to make sure I'm understanding things, you do mean up to n=900, 
all terms are either of the form a(n zeros)b , 1(n-k zeros)a(k zeros)b or 
1(n zeros)ab??

--------------------------------------------------
From: <israel at math.ubc.ca>
Sent: Friday, February 19, 2016 4:39 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Smallest prime with a substring of exactly n zeros

I now have a Maple program and b-file of 900 terms for A037053 (see
> draft edits), and indeed a(50) = 
> 10000000000000000000000000000000000000000000000500007. Up to n=900, all 
> terms are either of the form a(n zeros)b or 1(n-k zeros)a(k zeros)b, the 
> maximum k being 109 for A037053(808).
>
> Cheers,
> Robert
>