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A146310 Good approximation to the 10^n-th lower twin prime.

%I A146310 #5 Oct 28 2021 13:09:16
%S A146310 100,3380,75610,1257632,18456351,252177334,3285912624,41374714817,
%T A146310 507584081641,6100475249386,72109024427766,840671492062887,
%U A146310 9687559620379066,110531285543842366,1250315111094881329
%N A146310 Good approximation to the 10^n-th lower twin prime.
%C A146310 a(10) = 6100475249386 has relative 0.000000698 error from the actual value 6100479510551.
%H A146310 Cino Hilliard, <a href="http://groups.google.com/group/sumprimes/web/sumtwinsfirstn-c">Counting and summing primes</a>
%H A146310 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/twins/t2_0000.htm">Enumeration of twin primes less than 1e16</a>
%F A146310 Pi2(n) = number of twin primes <= n.
%F A146310 Twinpi(n) = number of twin prime pairs < n
%F A146310 Li_2(n)=intnum(t=2,n,2*c_2/log(t)^2)
%F A146310 The relationship n = Pi2(twinpi(n)) is used with a bisection routine where
%F A146310 Pi2(n) is the Hardy-Littlewood integral approximation for number of twin
%F A146310 primes
%o A146310 (PARI) g(n) = {
%o A146310 print1(floor(twinx2(10)),",");
%o A146310 for(x=2,n,y=twinx(10^x);print1(floor(y)","))
%o A146310 }
%o A146310 twinx(n) =
%o A146310 {
%o A146310 local(r1,r2,r,est);
%o A146310 r1 = n;
%o A146310 r2 = n*n;
%o A146310 for(x=1,100,
%o A146310 r=(r1+r2)/2.;
%o A146310 /*Hardy-Littlewood integral approximation for pi_2(x).*/
%o A146310 est = Li_2(r);
%o A146310 if(est <= n,r1=r,r2=r);
%o A146310 );
%o A146310 r;
%o A146310 }
%o A146310 twinx2(n) =
%o A146310 {
%o A146310 local(x,tx,r1,r2,r,pw,b,e,est);
%o A146310 if(n==1,return(3));
%o A146310 b=10;
%o A146310 pw=log(n)/log(b);
%o A146310 m=pw+1;
%o A146310 r1 = 0;
%o A146310 r2 = 7.213;
%o A146310 for(x=1,100,
%o A146310 r=(r1+r2)/2;
%o A146310 est = b^(m+r);
%o A146310 tx = Li_2(est);
%o A146310 if(tx <= b^pw,r1=r,r2=r);
%o A146310 );
%o A146310 est;
%o A146310 }
%o A146310 Li_2(x)=intnum(t=2,x,2*0.660161815846869573927812110014555778432623/log(t)^2)
%K A146310 nonn
%O A146310 1,1
%A A146310 _Cino Hilliard_, Oct 29 2008