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A052435 a(n) = round(li(n) - pi(n)), where li is the logarithmic integral and pi(x) is the number of primes <= x.

%I A052435 #42 Feb 16 2025 08:32:42
%S A052435 0,0,1,1,1,1,1,2,2,2,2,1,2,2,3,2,2,2,2,2,3,2,2,3,3,3,3,3,3,2,3,3,3,3,
%T A052435 4,3,3,4,4,3,3,3,3,3,3,3,3,3,3,4,4,3,3,4,4,4,4,4,4,3,3,4,4,4,4,4,4,4,
%U A052435 4,4,4,3,3,4,4,4,4,3,4,4,4,3,4,4,4,4,4,4,4,4,4,5,5,5,5,4,5,5,5,4,5,4
%N A052435 a(n) = round(li(n) - pi(n)), where li is the logarithmic integral and pi(x) is the number of primes <= x.
%C A052435 Eventually contains negative terms!
%C A052435 The logarithmic integral is the "American" version starting at 0.
%C A052435 The first crossover (P. Demichel) is expected to be around 1.397162914*10^316. - _Daniel Forgues_, Oct 29 2011
%H A052435 Harry J. Smith, <a href="/A052435/b052435.txt">Table of n, a(n) for n = 2..20000</a>
%H A052435 C. Caldwell, <a href="http://www.utm.edu/research/primes/howmany.shtml#hist">How many primes are there?</a>
%H A052435 Patrick Demichel, <a href="http://web.archive.org/web/20060908033007/http://demichel.net/patrick/li_crossover_pi.pdf">The prime counting function and related subjects</a>, April 05, 2005, 75 pages.
%H A052435 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>
%H A052435 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>
%H A052435 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SkewesNumber.html">Skewes Number</a>
%t A052435 Table[Round[LogIntegral[x]-PrimePi[x]], {x,2,100}]
%o A052435 (PARI) a(n)=round(real(-eint1(-log(n)))-primepi(n)) \\ _Charles R Greathouse IV_, Oct 28 2011
%o A052435 (Magma) [Round(LogIntegral(n) - #PrimesUpTo(n)): n in [2..105]]; // _G. C. Greubel_, May 17 2019
%o A052435 (Sage) [round(li(n) - prime_pi(n)) for n in (2..105)] # _G. C. Greubel_, May 17 2019
%Y A052435 Cf. A000720, A052434, A282870, A359145.
%K A052435 sign,look
%O A052435 2,8
%A A052435 _Eric W. Weisstein_