A226744 -id:A226744 - cp's OEIS frontend

A193257 Floor((10^n)/(log(10^n) - 1)).

7, 27, 169, 1217, 9512, 78030, 661458, 5740303, 50701542, 454011971, 4110416300, 37550193649, 345618860220, 3201414635780, 29816233849000, 279007258230819, 2621647966812031, 24723998785919976, 233922961602470390, 2219671974013732243

Offset: 1

Arkadiusz Wesolowski, Jul 19 2011

nonn

lim n -> infinity (log(n) - n/pi(n)) = 1, where pi(n) is the prime counting function.

			a(2) = 27 because (10^2)/(log(10^2) - 1) = 27.7379415786....

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

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..200
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 A226744.

Cf. A058289, A006880, A057834, A000720.

[Floor(10^n/(Log(10^n)-1)) : n in [1..20]]

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

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

a(n)=10^n\(n*log(10)-1) \\ Charles R Greathouse IV, Jul 30 2011

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

A226945 Integer nearest f(10^n), where f(x) = Sum of ( mu(k) * H(k)/k^(3/2) * Integral Log(x^(1/k)) ) for k = 1 to infinity, where H(k) is the harmonic number sum_{i=1..k} 1/i.

Original entry on oeis.org

4, 25, 168, 1226, 9585, 78521, 664652, 5761512, 50847348, 455050385, 4118051652, 37607908133, 346065524108, 3204941711340, 29844570436484, 279238341185832, 2623557156537070, 24739954282695698, 234057667295619287, 2220819602542218793

Offset: 1

Arkadiusz Wesolowski, Aug 31 2013

nonn

The sequence gives exactly the values of pi(10^n) for n = 1 to 3.

A228724 gives the difference between A006880 and this sequence.

David Baugh, Table of n, a(n) for n = 1..100
Chris K. Caldwell, How Many Primes Are There?
Eric Weisstein's World of Mathematics, Prime Counting Function
Eric Weisstein's World of Mathematics, Prime Number Theorem

Cf. A006880, A057834, A226744, A057754, A190802, A057793, A201542, A228724.

f[n_Integer] := Sum[N[MoebiusMu[k]*HarmonicNumber[k]/k^(3/2)*LogIntegral[n^(1/k)], 50], {k, 5!}]; Table[Round[f[10^n]], {n, 20}]

cp's OEIS Frontend

A193257 Floor((10^n)/(log(10^n) - 1)).

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