A331020 Values k for successive maximal records of the function A defined as A(prime(k)) = log(prime(k)) - prime(k)/Pi(prime(k)) where Pi(prime(k)) is number of primes <= prime(k).

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 15, 18, 21, 27, 28, 29, 30, 46, 61, 91, 121, 180, 184, 185, 186, 188, 189, 214, 216, 217, 257, 258, 775, 832, 1217, 1225, 1227, 1269, 1270, 1846, 1847, 2682, 2683, 2684, 2685, 2686, 2688

Offset: 1

Author

Artur Jasinski, Jan 07 2020

Keywords

• nonn
• fini
• full

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.

Examples

			   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
		

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.

Crossrefs

Cf. A000720, A228211.

Programs

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