1,1

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

Cino Hilliard, Counting and summing primes

T. R. Nicely, Enumeration of twin primes less than 1e16

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

(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)

Sequence in context: A112889 A118490 A165518 this_sequence A117685 A091134 A017816

Adjacent sequences: A146307 A146308 A146309 this_sequence A146311 A146312 A146313

nonn

Cino Hilliard (hillcino368(AT)hotmail.com), Oct 29 2008