OFFSET
1,2
COMMENTS
This sequence is finite and complete.
Chebyshev 1852, goes on to conclude that if we put Pi(x) = x/(log(x) - A(x)) has a limit as x -> +infinity, then a limit must be 1, not 1.08366 (A228211), as Legendre incorrectly conjectured in 1808.
R. Farhadian & R. Jakimczuk 2018 prove again that the function A tends to 1 when n tends to infinity.
A(prime(2688)) = A(24137) = -24137/2688 + log(24137) = 1.11196252139...
A(n) <= -(24137/2688) + log(24137) for all positive integers n.
LINKS
P. L. Chebyshev, Sur la totalité des nombres premiers inférieurs à une limite donnée, J. math. pures appl. 17, 1852 (in French).
R. Farhadian & R. Jakimczuk, One more disproof for the Legendre's conjecture regarding the prime counting function Pi(x), Notes on Number Theory and Discrete Mathematics, Vol. 24, 2018, No. 3, 84-91.
Eric Weisstein's World of Mathematics, Legendre's Constant.
EXAMPLE
n | a(n) | A(prime(a(n)))
---+------+---------------
1 | 1 | -1.306852819
2 | 2 | -0.401387711
3 | 3 | -0.057228754
4 | 4 | 0.195910149
5 | 5 | 0.197895272
6 | 6 | 0.398282690
7 | 7 | 0.404641915
8 | 8 | 0.569438979
9 | 9 | 0.579938660
10 | 11 | 0.615805386
MATHEMATICA
max = -2; aa = {}; Do[kk = Log[Prime[n]] - Prime[n]/PrimePi[Prime[n]];
If[kk > max, max = kk; AppendTo[aa, n]], {n, 1, 2700}]; aa
CROSSREFS
KEYWORD
nonn,fini,full,changed
AUTHOR
Artur Jasinski, Jan 07 2020
STATUS
approved