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URL: https://oeis.org/A190802

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A190802
Gauss' approximation for the number of primes below 10^n.
10
5, 29, 177, 1245, 9629, 78627, 664917, 5762208, 50849234, 455055614, 4118066400, 37607950280, 346065645809, 3204942065691, 29844571475287, 279238344248556, 2623557165610821, 24739954309690414, 234057667376222381, 2220819602783663483, 21127269486616126181, 201467286691248261497
OFFSET
1,1
COMMENTS
The offset logarithmic integral or Eulerian logarithmic integral Li(10^n)-Li(2), i.e., Integral_{t=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
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.
FORMULA
a(n) = round(Integral_{t=2..10^n} dt/log(t)).
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 *)
CROSSREFS
Sequence in context: A163611 A160906 A163073 * A139174 A290117 A153296
KEYWORD
nonn
AUTHOR
Nathaniel Johnston, May 25 2011
STATUS
approved