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