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

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, 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, 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

Offset: 2

Author

Eric W. Weisstein

Keywords

• sign
• look

Comments

Eventually contains negative terms!

The logarithmic integral is the "American" version starting at 0.

The first crossover (P. Demichel) is expected to be around 1.397162914*10^316. - Daniel Forgues, Oct 29 2011

Links

• Harry J. Smith, Table of n, a(n) for n = 2..20000
• C. Caldwell, How many primes are there?
• Patrick Demichel, The prime counting function and related subjects, April 05, 2005, 75 pages.
• Eric Weisstein's World of Mathematics, Prime Counting Function
• Eric Weisstein's World of Mathematics, Logarithmic Integral
• Eric Weisstein's World of Mathematics, Skewes Number

Crossrefs

Cf. A000720, A052434, A282870, A359145.

Programs

Magma

[Round(LogIntegral(n) - #PrimesUpTo(n)): n in [2..105]]; // G. C. Greubel, May 17 2019

Mathematica

Table[Round[LogIntegral[x]-PrimePi[x]], {x,2,100}]

PARI

a(n)=round(real(-eint1(-log(n)))-primepi(n)) \\ Charles R Greathouse IV, Oct 28 2011

Sage

[round(li(n) - prime_pi(n)) for n in (2..105)] # G. C. Greubel, May 17 2019