A128910 Similar to A057835 except using K * X / log(X), K=1.022.

0, 3, 20, 119, 715, 4523, 30509, 213343, 1530983, 11203550, 83064263, 620498643, 4643259527, 34592032908, 254639722327, 1832740718223, 12680919388801, 81678704122892, 452951221016511, 1574800035301944, 8395299939524712, 282240813012897282, 4457697545906326118, 58106920364272792945, 693274802905577732102, 7864635685729658131835

Offset

1,2

Comments

This is an improvement over the classic X / log(X) approximation in the range many people work with.

pi(x), R(x), and li(x) are all asymptotically x/log x + x/log^2 x + O(x/log^3 x), so this approximation is good around exp(1/.022) ≈ 5 * 10^19. Asymptotically the best value for K would be 1. - Charles R Greathouse IV, Aug 18 2022

Links

Table of n, a(n) for n=1..26.

Formula

a(n) = abs(round(1.022*10^n/log(10^n)) - pi(10^n)). - Charles R Greathouse IV, Mar 22 2015

a(n) ~ 10^n/kn with k = 104.6629.... - Charles R Greathouse IV, Mar 22 2015

Example

a(10)=11203550 via abs (455,052,511 - 443,848,961).

Mathematica

Table[ PrimePi[10^n] - Round[N[1.022*10^n/Log[10^n]]], {n, 23}] (* and absolute value thereof (orig entries 21-23 <0); courtesy of Robert G. Wilson v *)

Prog

(PARI) a(n) = abs(round(1.022*10^n/log(10^n)) - primepi(10^n)) \\ Charles R Greathouse IV, Mar 22 2015

Crossrefs

Cf. A057835, A006880.

Sequence in context: A275796 A164535 A001652 * A037788 A037669 A228066

Adjacent sequences: A128907 A128908 A128909 * A128911 A128912 A128913

Keyword

nonn

Author

Bill McEachen, Apr 23 2007

Status

approved