A202766 Floor( 10^n / sum(k=3..10^n, 1/k ) ).

6, 27, 167, 1206, 9442, 77563, 658097, 5714972, 50503822, 452425909, 4097411586, 37441633014, 344698955565, 3193520274110, 29747746198318, 278407464679282, 2616351626277085, 24676888631241563, 233501199663256017, 2215874110986269907

Offset: 1

Author

Arkadiusz Wesolowski, Dec 23 2011

Keywords

• nonn

Comments

n/(Sum_{k=3..n} 1/k) is a better approximation to pi(n) than Gauss' Li(n) for 15 < n < 2803.

Examples

			a(2) = 27 because (10^2)/(Sum_{k=3..100} 1/k) = 27.1195448585....
		

References

• Wacław Sierpiński, Co wiemy, a czego nie wiemy o liczbach pierwszych. Warsaw: PZWS, 1961, p. 21.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Prime Counting Function
• Eric Weisstein's World of Mathematics, Prime Number Theorem

Crossrefs

Cf. A000720, A006880, A193257.

Programs

Mathematica

lst = {}; Do[AppendTo[lst, Floor[10^n/(NIntegrate[(1 - x^10^n)/(1 - x), {x, 0, 1}, WorkingPrecision -> 20] - 1.5)]], {n, 13}]; lst

Formula

a(n) = floor((10^n)/(Sum_{k=3..10^n} 1/k)).

a(n) ~ 10^n/(log(10^n) + gamma - 3/2).