A193257 -id:A193257 - cp's OEIS frontend

A226744 Round((10^n)/(log(10^n) - 1)).

8, 28, 169, 1218, 9512, 78030, 661459, 5740304, 50701542, 454011971, 4110416301, 37550193650, 345618860221, 3201414635781, 29816233849001, 279007258230820, 2621647966812031, 24723998785919976, 233922961602470391, 2219671974013732243

Offset: 1

Arkadiusz Wesolowski, Aug 31 2013

nonn

			a(2) = 28 because (10^2)/(log(10^2) - 1) = 27.7379415786....

A. M. Legendre, Essai sur la Théorie des Nombres, Paris: Duprat, 1808.

Eric Weisstein's World of Mathematics, Legendre's Constant
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Prime Number Theorem

Another version of A193257.

Cf. A058289, A006880, A057834, A000720.

Table[Round[10^n/(Log[10^n] - 1)], {n, 20}]

for(n=1, 20, print1(round(10^n/(log(10^n)-1)), ", "));

a(n) = round((10^n)/(log(10^n) - 1)).

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

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Dec 23 2011

nonn

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

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

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

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

Cf. A000720, A006880, A193257.

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

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

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

cp's OEIS Frontend

A226744 Round((10^n)/(log(10^n) - 1)).

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