sequence of Zak Seidov needs extending

[Max]

Probabilistic arguments indicate that there are a finite number of terms in this sequence:
the probability of n being prime is roughly 1/ln(n)
the probability of n being palindromic is roughly 1/sqrt(n)
assuming these two properties are indepenedent, the probability of n being a palindromic prime is roughly 1/(sqrt(n)*ln(n))
let m be n-th prime (i.e., pi(m)=n), so m is roughly n*ln(n) and the probability of m being palindromic is roughly 1/sqrt(n*ln(n))
Hence, the probability of m belonging A103359 is roughly 1/(n*ln(n)^(3/2)) and integral(1/(n*ln(n)^(3/2)),n=2..oo) is finite.

So there is a great chance that 3,5,11 are actually all such numbers.

Alex, I wonder how did you get this far. Did you use advanced methods (like the Meissel-Lehmer method) to compute pi(x) or nthprime(x)?

Regards,
Max

Alex Healy wrote:
> I also threw together a quick program for this, and can't seem to find any
> other terms less than (2.5)*10^12.  (That's just where my program was when I
> woke up this morning.)
> 
> For what it's worth, the first four terms of the same sequence in binary
> (i.e. requiring that the primes be palindromic in base two) are:
> 
> 5, 17, 127, 296713
> 
> and there don't seem to be any others less than 10^11.
> 
> Alex 
> 
> 
>>-----Original Message-----
>>From: Jud McCranie [mailto:j.mccranie at adelphia.net] 
>>Sent: Thursday, February 03, 2005 12:52 AM
>>To: njas at research.att.com
>>Cc: seqfan at ext.jussieu.fr; zakseidov at yahoo.com
>>Subject: Re: sequence of Zak Seidov needs extending
>>
>>At 11:00 PM 2/2/2005, N. J. A. Sloane wrote:
>>
>>>Dear Seqfans, Zak submitted this, but I can't use it without 
>>
>>at least 
>>
>>>one more term. Can anyone extend it?
>>>
>>>
>>>%I A103359
>>>%S A103359 3,5,11
>>>%N A103359 Prime palindromic n such that pi(n) [A000720] is 
>>
>>prime palindromic.
>>
>>>%C A103359 No further terms with n less than 3000000. 
>>
>>Palindromic pi(n) 
>>
>>>of palindromic n A103357,A103358.
>>>%e A103359 pi(3)=2, pi(5)=3,pi(11)=5
>>>%Y A103359 Cf. A103357, A103358.
>>>%O A103359 0,1
>>>%K A103359 more,nonn,base,bref
>>>%A A103359 Zak Seidov (zakseidov(AT)yahoo.com), Feb 02 2005
>>
>>My quick and dirty program doesn't find any more terms < 
>>10^10.  Tomorrow I can modify it to go higher.
>>
>>More terms are going to be hard to find.
>>
>>pi(143787341) =  8114118, a palindrome, but not prime - is 
>>the closest I've found. (143787341 is a palindromic prime).
>>
>>You might have to relax the condition that pi(n) be a prime, 
>>and just require it to be palindromic, while n is a palindromic prime.
>>
>>
> 
> 
> 
>