A223167 Difference between nearest integer to (Li(10^n)-Li(3)) and pi(10^n), where Li(10^n)-Li(3) = integral(3.. 10^n, dt/log(t)) (A223166) and pi(10^n) = number of primes <= 10^n (A006880).

0, 3, 7, 15, 36, 127, 337, 752, 1699, 3101, 11585, 38261, 108969, 314888, 1052616, 3214630, 7956587, 21949553, 99877773, 222744641, 597394252, 1932355206, 7250186214, 17146907276, 55160980937, 155891678119, 508666658004, 1427745660372

Offset: 1

Author

Vladimir Pletser, Mar 16 2013

Keywords

• nonn

Comments

As Li(3)= 2.163588..., A057752(n)-a(n) = 2, except for n =3, 6, 10, 11, 15, 20 where A057752(n)-a(n)= 3.

This sequence yields an even better average relative difference than Gauss's approximation (A106313), i.e., Average(a(n)/pi(10^n)) = 7.4969...*10^-3 for 1<=n<=24, compared to Average(A057752(n)/pi(10^n)) = 3.2486...*10^-2 and Average(A106313(n)/pi(10^n)) = 2.0116...*10^-2, showing that, when using the logarithmic integral, Li(10^n)-Li(3) (A223166) gives a better approximation to pi(10^n) than Li(10^n)-Li(2) (A190802) and than Li(10^n) (A057754).

Links

• Eric Weisstein's World of Mathematics, Prime Counting Function
• Eric Weisstein's World of Mathematics, Logarithmic Integral

Crossrefs

Cf. A057754, A057752, A006880, A190802, A106313.

Programs

Mathematica

a[n_] := Round[LogIntegral[10^n] - LogIntegral[3]] - PrimePi[10^n]; Table[a[n], {n, 1, 14}]

PARI

a(n)=round(eint1(-log(3))-eint1(-n*log(10)))-primepi(10^n) \\ Charles R Greathouse IV, May 03 2013

Formula

a(n) = A223166(n) - A006880(n).

Extensions

Terms a(25)-a(28) obtained using A006880. - Eduard Roure Perdices, Apr 14 2021