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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).

%I A331020 #36 Dec 19 2024 21:17:15
%S A331020 1,2,3,4,5,6,7,8,9,11,14,15,18,21,27,28,29,30,46,61,91,121,180,184,
%T A331020 185,186,188,189,214,216,217,257,258,775,832,1217,1225,1227,1269,1270,
%U A331020 1846,1847,2682,2683,2684,2685,2686,2688
%N 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).
%C A331020 This sequence is finite and complete.
%C A331020 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.
%C A331020 R. Farhadian & R. Jakimczuk 2018 prove again that the function A tends to 1 when n tends to infinity.
%C A331020 A(prime(2688)) = A(24137) = -24137/2688 + log(24137) = 1.11196252139...
%C A331020 A(n) <= -(24137/2688) + log(24137) for all positive integers n.
%H A331020 P. L. Chebyshev, <a href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A18_0.pdf">Sur la totalité des nombres premiers inférieurs à une limite donnée</a>, J. math. pures appl. 17, 1852 (in French).
%H A331020 R. Farhadian & R. Jakimczuk, <a href="https://doi.org/10.7546/nntdm.2018.24.3.84-91">One more disproof for the Legendre's conjecture regarding the prime counting function Pi(x)</a>, Notes on Number Theory and Discrete Mathematics, Vol. 24, 2018, No. 3, 84-91.
%H A331020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendresConstant.html">Legendre's Constant</a>.
%e A331020    n | a(n) | A(prime(a(n)))
%e A331020   ---+------+---------------
%e A331020    1 |    1 | -1.306852819
%e A331020    2 |    2 | -0.401387711
%e A331020    3 |    3 | -0.057228754
%e A331020    4 |    4 |  0.195910149
%e A331020    5 |    5 |  0.197895272
%e A331020    6 |    6 |  0.398282690
%e A331020    7 |    7 |  0.404641915
%e A331020    8 |    8 |  0.569438979
%e A331020    9 |    9 |  0.579938660
%e A331020   10 |   11 |  0.615805386
%t A331020 max = -2; aa = {}; Do[kk = Log[Prime[n]] - Prime[n]/PrimePi[Prime[n]];
%t A331020 If[kk > max, max = kk; AppendTo[aa, n]], {n, 1, 2700}]; aa
%Y A331020 Cf. A000720, A228211.
%K A331020 nonn,fini,full
%O A331020 1,2
%A A331020 _Artur Jasinski_, Jan 07 2020