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

100, 3380, 75610, 1257632, 18456351, 252177334, 3285912624, 41374714817, 507584081641, 6100475249386, 72109024427766, 840671492062887, 9687559620379066, 110531285543842366, 1250315111094881329

Offset: 1

Author

Cino Hilliard, Oct 29 2008

Keywords

• nonn

Comments

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

Links

• Cino Hilliard, Counting and summing primes
• Thomas R. Nicely, Enumeration of twin primes less than 1e16

Programs

PARI

g(n) = {
print1(floor(twinx2(10)),",");
for(x=2,n,y=twinx(10^x);print1(floor(y)","))
}
twinx(n) =
{
local(r1,r2,r,est);
r1 = n;
r2 = n*n;
for(x=1,100,
r=(r1+r2)/2.;
/*Hardy-Littlewood integral approximation for pi_2(x).*/
est = Li_2(r);
if(est <= n,r1=r,r2=r);
);
r;
}
twinx2(n) =
{
local(x,tx,r1,r2,r,pw,b,e,est);
if(n==1,return(3));
b=10;
pw=log(n)/log(b);
m=pw+1;
r1 = 0;
r2 = 7.213;
for(x=1,100,
r=(r1+r2)/2;
est = b^(m+r);
tx = Li_2(est);
if(tx <= b^pw,r1=r,r2=r);
);
est;
}
Li_2(x)=intnum(t=2,x,2*0.660161815846869573927812110014555778432623/log(t)^2)

Formula

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

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

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

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

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

primes