A152051 Hardy-Littlewood approximation to the number of twin primes less than 10^n.

5, 14, 46, 214, 1249, 8248, 58754, 440368, 3425308, 27411417, 224368865, 1870559867, 15834598305, 135780264894, 1177208491861, 10304192554496, 90948833260990, 808675901493606, 7237518062753712, 65154265428712141

Offset: 1

Author

Cino Hilliard, Nov 22 2008

Keywords

• nonn

Comments

Another good approximation to the number of twin primes < 10^n is the sum of twin primes < 10^(n/2)/4. For example Pi2(10^16) = 10304185697298.

SumPi2(10^8)/4 = 10301443659233 for an error of 0.0000266. However, the Hardy-Littlewood approximation is superior giving an error of -0.000000665.

Programs

PARI

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

Formula

C_2 = 0.660161815846869573927812110014555778432623. Li_2(x) = 2*C_2*integral(t=2..x,dt/log(t)^2)