A202766 - OEIS

%I #31 Feb 16 2025 08:33:16

%S 6,27,167,1206,9442,77563,658097,5714972,50503822,452425909,

%T 4097411586,37441633014,344698955565,3193520274110,29747746198318,

%U 278407464679282,2616351626277085,24676888631241563,233501199663256017,2215874110986269907

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

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

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

%H Arkadiusz Wesolowski, <a href="/A202766/b202766.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeNumberTheorem.html">Prime Number Theorem</a>

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

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

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

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

%Y Cf. A000720, A006880, A193257.

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Dec 23 2011