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%I #8 May 25 2018 08:28:03
%S 2,1051,1063,1069,1097,1103,1109,1123,1129,1303,1307,1321,1327,1619,
%T 1621,1627,2399,2447,2477,2719,2731,2753,2801,2803,3929,3931,3947,
%U 4273,4289,4297,5851,5857,5861,5867,5869,5881,6367,6373,6379,9433,9437,9439
%N x such that pi(x)/li(x) is greater than it is for all smaller x > 1.5.
%C This will be a very long but finite sequence, since pi(x)/li(x) will exceed unity for some very large values of x (as Littlewood first showed) but then will asymptotically tend to unity by the prime number theorem. One large but unknown element of the sequence will be the smallest x for which pi(x)>li(x).
%e For 1.5<x<2, li(x)>0 and pi(x)=0, so pi(x)/li(x)=0. a(1)=2 because at x=2, pi(x)/li(x) has its increase, to 1/li(2)=0.9567878442. a(2)=1051 because x=1051 gives the next time pi(x)/li(x) gives a higher value, 177/Li(1051)=0.956932676.
%p with(numtheory): Digits:=50; s:=0: for n from 1 to 10000 do if (evalf(n/Li(ithprime(n)))>s) then s:=evalf(n/Li(ithprime(n))): print(ithprime(n)) else s:=s end if end do;
%K nonn,fini
%O 1,1
%A _Don N. Page_, Oct 24 2005
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