cp's OEIS Frontend

A228211 Decimal expansion of Legendre's constant (incorrect, the true value is 1, as in A000007).

%I A228211 #83 Apr 19 2025 06:11:09
%S A228211 1,0,8,3,6,6
%N A228211 Decimal expansion of Legendre's constant (incorrect, the true value is 1, as in A000007).
%C A228211 Included in accordance with the OEIS policy of listing incorrect but published sequences. The correct value of this constant is 1, by the prime number theorem pi(x) ~ li(x) = x/(log(x) - 1 - 1/log(x) + O(1/log^2(x))), where li is the logarithmic integral.
%C A228211 Before the prime number theorem was proved, it was believed that there was a constant A not equal to 1 that needed to be inserted in the formula pi(n) ~ n/(log(n) - A) to make it more precise. This number was Adrien-Marie Legendre's guess.
%C A228211 Panaitopol proved that x/(log(x) - A), where A is this constant, is an upper bound for pi(x) when x > 10^6. - _John W. Nicholson_, Feb 26 2018
%D A228211 Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 80.
%D A228211 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 163.
%D A228211 Hans Riesel, Prime Numbers and Computer Methods for Factorization. New York: Springer (1994): 41 - 43.
%H A228211 Kevin Brown, <a href="http://www.mathpages.com/home/kmath032.htm">Legendre's Prime Number Conjecture</a>.
%H A228211 Alexei Kourbatov, <a href="https://arxiv.org/abs/1610.03340">On the distribution of maximal gaps between primes in residue classes</a>, arXiv:1610.03340 [math.NT], 2016-2017.
%H A228211 Laurenţiu Panaitopol, <a href="http://dx.doi.org/10.7153/mia-02-29">Several Approximations of pi(x)</a>, Math. Ineq. Appl. 2(1999), 317-324.
%H A228211 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendresConstant.html">Legendre's Constant</a>.
%F A228211 Believed at one time to be lim_{n -> oo} A(n) in pi(n) = n/(log(n) - A(n)).
%e A228211 A = 1.08366.
%Y A228211 Cf. A000007.
%K A228211 nonn,cons,fini,full
%O A228211 1,3
%A A228211 _Alonso del Arte_, Nov 02 2013
%E A228211 Edited by _N. J. A. Sloane_, Nov 13 2014