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A282870 a(n) = floor( Li(n) - pi(n) ).

%I A282870 #25 Feb 07 2023 17:11:11
%S A282870 0,0,0,0,1,0,1,1,2,1,2,1,1,2,2,1,2,1,1,2,2,1,2,2,2,3,3,2,3,2,2,2,3,3,
%T A282870 3,3,3,3,3,3,3,2,2,3,3,2,2,3,3,3,3,3,3,3,3,4,4,3,3,3,3,3,3,4,4,3,3,4,
%U A282870 4,3,3,3,3,3,3,3,4,3,3,3,4,3,3,3,4,4,4,3,3,4,4,4,4,5,5,4,4,4,5,4,4,3,3,4,4
%N A282870 a(n) = floor( Li(n) - pi(n) ).
%C A282870 Li(x) is the logarithmic integral of x.
%C A282870 pi(x) is the number of primes less than or equal to x, A000720(x).
%C A282870 "The Riemann hypothesis is an assertion about the size of the error term in the prime number theorem, namely, that pi(x) = li(x)+O(x^(1/2+epsilon))", see Nathanson, page 323.
%D A282870 Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, 2000
%F A282870 a(n) = A047783(n) - A000720(n).
%p A282870 a:= n-> floor(evalf(Li(n)))-numtheory[pi](n):
%p A282870 seq(a(n), n=2..120);  # _Alois P. Heinz_, Feb 23 2017
%t A282870 iend = 100;
%t A282870 For[x = 1, x <= iend, x++,
%t A282870 a[x] = N[LogIntegral[x] - PrimePi[x]]]; t =
%t A282870 Table[Floor[a[i]], {i, 2, iend}]; Print[t]
%t A282870 Table[Floor[LogIntegral[n] - PrimePi[n]], {n, 2, 110}] (* _G. C. Greubel_, May 17 2019 *)
%o A282870 (PARI) vector(110, n, n++; floor(real(-eint1(-log(n))) - primepi(n)) ) \\ _G. C. Greubel_, May 17 2019
%o A282870 (Magma) [Floor(LogIntegral(n) - #PrimesUpTo(n)): n in [2..110]]; // _G. C. Greubel_, May 17 2019
%o A282870 (Sage) [floor(li(n) - prime_pi(n)) for n in (2..110)] # _G. C. Greubel_, May 17 2019
%Y A282870 Cf. A000720, A047783, A052435, A359145.
%K A282870 sign
%O A282870 2,9
%A A282870 _David S. Newman_, Feb 23 2017