

A228211


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


1




OFFSET

1,3


COMMENTS

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.
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 AdrienMarie Legendre's guess.
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


REFERENCES

Hans Riesel, Prime Numbers and Computer Methods for Factorization. New York: Springer (1994): 41  43.


LINKS

Table of n, a(n) for n=1..6.
Kevin Brown, Legendre's Prime Number Conjecture.
Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 20162017.
LaurenÅ£iu Panaitopol, Several Approximations of pi(x), Math. Ineq. Appl. 2(1999), 317324.
Eric Weisstein's World of Mathematics, "Legendre's Constant".


FORMULA

Believed at one time to be lim_{n > infinity} A(n) in pi(n) = n/(log(n)  A(n)).


EXAMPLE

A = 1.08366.


CROSSREFS

Cf. A000007.
Sequence in context: A304580 A304583 A104697 * A010522 A197332 A258991
Adjacent sequences: A228208 A228209 A228210 * A228212 A228213 A228214


KEYWORD

nonn,cons,fini,full


AUTHOR

Alonso del Arte, Nov 02 2013


EXTENSIONS

Edited by N. J. A. Sloane, Nov 13 2014


STATUS

approved



