A146310 - OEIS

%I #5 Oct 28 2021 13:09:16

%S 100,3380,75610,1257632,18456351,252177334,3285912624,41374714817,

%T 507584081641,6100475249386,72109024427766,840671492062887,

%U 9687559620379066,110531285543842366,1250315111094881329

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

%C a(10) = 6100475249386 has relative 0.000000698 error from the actual value 6100479510551.

%H Cino Hilliard, <a href="http://groups.google.com/group/sumprimes/web/sumtwinsfirstn-c">Counting and summing primes</a>

%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/twins/t2_0000.htm">Enumeration of twin primes less than 1e16</a>

%F Pi2(n) = number of twin primes <= n.

%F Twinpi(n) = number of twin prime pairs < n

%F Li_2(n)=intnum(t=2,n,2*c_2/log(t)^2)

%F The relationship n = Pi2(twinpi(n)) is used with a bisection routine where

%F Pi2(n) is the Hardy-Littlewood integral approximation for number of twin

%F primes

%o (PARI) g(n) = {

%o print1(floor(twinx2(10)),",");

%o for(x=2,n,y=twinx(10^x);print1(floor(y)","))

%o }

%o twinx(n) =

%o {

%o local(r1,r2,r,est);

%o r1 = n;

%o r2 = n*n;

%o for(x=1,100,

%o r=(r1+r2)/2.;

%o /*Hardy-Littlewood integral approximation for pi_2(x).*/

%o est = Li_2(r);

%o if(est <= n,r1=r,r2=r);

%o );

%o r;

%o }

%o twinx2(n) =

%o {

%o local(x,tx,r1,r2,r,pw,b,e,est);

%o if(n==1,return(3));

%o b=10;

%o pw=log(n)/log(b);

%o m=pw+1;

%o r1 = 0;

%o r2 = 7.213;

%o for(x=1,100,

%o r=(r1+r2)/2;

%o est = b^(m+r);

%o tx = Li_2(est);

%o if(tx <= b^pw,r1=r,r2=r);

%o );

%o est;

%o }

%o Li_2(x)=intnum(t=2,x,2*0.660161815846869573927812110014555778432623/log(t)^2)

%K nonn

%O 1,1

%A _Cino Hilliard_, Oct 29 2008