A190802 Gauss' approximation for the number of primes below 10^n.

5, 29, 177, 1245, 9629, 78627, 664917, 5762208, 50849234, 455055614, 4118066400, 37607950280, 346065645809, 3204942065691, 29844571475287, 279238344248556, 2623557165610821, 24739954309690414, 234057667376222381, 2220819602783663483

Offset: 1

Author

Nathaniel Johnston, May 25 2011

Keywords

• nonn

Comments

The offset logarithmic integral or Eulerian logarithmic integral Li(10^n)-Li(2), i.e., integral(2..x, dt/log(t)), appears in Gauss’s formula for counting prime numbers < 10^n and is sometimes referred to as the "European" definition. - Vladimir Pletser, Mar 17 2013

References

• Jonathan Borwein, David H. Bailey, "Mathematics by Experiment", A. K. Peters, 2004, p. 65 (Table 2.2).

Links

• Vladimir Pletser, Table of n, a(n) for n = 1..500
• Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 1 p. 6.

Crossrefs

Cf. A006880, A106313.

Programs

Maple

seq(round(evalf(integrate(1/log(t),t=2..10^n))), n=1..21);

Mathematica

Table[Round[Integrate[1/Log[t],{t,2,10^n}]],{n,20}] (* James C. McMahon, Feb 06 2024 *)

Formula

a(n) = round(integral(dt/log(t),t=2..10^n)).