[seqfan] Re: more digits of sum over squared inverse twin primes

[Faride Firoozbakht]

  I submitted the following comment.

  We can show that a(9)=6, a(10)=5 and a(11) is in the set {7, 8, 9}.

  Proof:

  s1 = 0.237251776576249072... is the sum up to prime(499,000,000)
  s2 = 0.237251776576250009... is the sum up to prime(500,000,000).
  By using the fact that number of twin primes between the first 10^6*n primes
  and the first 10^6*(n+1) primes is decreasing (up to first 2*10^9 primes), we
  conclude that the sum up to prime(2000,000,000)is less than s2+1500*(s2-s1).
  But since s2-s1 < 10^(-15) so the sum up to prime(2*10^9) is less than
  s2 + 1.5*10^(-12) = 0.237251776576250009... + 1.5*10^(-12) =
  0.237251776577550009... .
  Hence the constant c is less than  0.237251776577550009... +
  lim(sum(1/k^2,{k, prime(2,000,000,001), n}, n -> infinity)
  < 0.237251776577550009... + 2.12514*10^(-11)
  < 0.237251776598801409.
  So we have 0.237251776576250009 < c < 0.237251776598801409, hence a(9)=6,
  a(10)=5 and a(11) is in the set {7, 8, 9}.

  I guess that a(11)=7.

  ---Farideh


Quoting Farideh Firoozbakht <f.firoozbakht at sci.ui.ac.ir>:

>
>> Richard Mathar
>> snapshots of results:
>> prime(900000)  0.23725177594...
>> prime(1290000) 0.23725177616...
>> ...
>> might test whether the 0.23725177 may turn to 0.23725178 later on.
>
>
> No, 0.23725177 may not turn to 0.23725178.
> Because,
>
> prime(100,000,000)     0.237251776574074...
> prime(110,000,000)     0.237251776574342...
> prime(120,000,000)     0.237251776574562...
> prime(130,000,000)     0.237251776574746...
>
> So the constant c is less than
>     0.237251776574747 + lim(sum(1/k^2,{k, prime(130,000,001), n}, n ->
> infinity)
>   < 0.237251776574747 + 3.72376*10^(-10) < 0.237251776947124
>
> Hence 0.237251776574746 < c < 0.237251776947124 and we conclude that the
> first nine terms of the sequence are:  2, 3, 7, 2, 5, 1, 7, 7, 6
>
> Rigards,
> Farideh
>
>
>
>
> Quoting Richard Mathar <mathar at strw.leidenuniv.nl>:
>
>>
>> The current entry of  0.237251058
>> http://research.att.com/~njas/sequences/A160910
>> seems to be incorrect in at least three decimal digits.
>> I guess it is defined as the sum over all 1/A001359(n)^2+1/A006512(n)^2.
>> Gathering twin primes up to prime(1070000) the constant
>> is at least 0.23725177606 .
>> Someone with some spare computer cycles---not the cycles that spin the
>> computer and the earth around the sun in a year but the other,
>> quicker ones---
>> might test whether the 0.23725177 may turn to 0.23725178 later on.
>>
>> Richard Mathar
>> snapshots of results:
>> prime(900000)  0.23725177594...
>> prime(1290000) 0.23725177616...
>>
>> Digits := 40 ;
>> x := 0.0 ;
>> for n from 1 do
>>         p := ithprime(n) ;
>>         if isprime(p+2) then
>>                 x := evalf(x+1/p^2+1/(p+2)^2) ;
>>         fi;
>>         if n mod 10000 = 0 then
>>         print(n,x) ;
>>         fi;
>> od:
>>
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/

>
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