A359145 -id:A359145 - cp's OEIS frontend

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

Eric W. Weisstein

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

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

Cf. A000720, A052434, A282870, A359145.

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

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

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

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

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

Original entry on oeis.org

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, 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, 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

Offset: 2

David S. Newman, Feb 23 2017

sign

Li(x) is the logarithmic integral of x.
pi(x) is the number of primes less than or equal to x, A000720(x).
"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.

Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, 2000

Cf. A000720, A047783, A052435, A359145.

[Floor(LogIntegral(n) - #PrimesUpTo(n)): n in [2..110]]; // G. C. Greubel, May 17 2019

a:= n-> floor(evalf(Li(n)))-numtheory[pi](n):
seq(a(n), n=2..120);  # Alois P. Heinz, Feb 23 2017

iend = 100;
For[x = 1, x <= iend, x++,
a[x] = N[LogIntegral[x] - PrimePi[x]]]; t =
Table[Floor[a[i]], {i, 2, iend}]; Print[t]
Table[Floor[LogIntegral[n] - PrimePi[n]], {n, 2, 110}] (* G. C. Greubel, May 17 2019 *)

vector(110, n, n++; floor(real(-eint1(-log(n))) - primepi(n)) ) \\ G. C. Greubel, May 17 2019

[floor(li(n) - prime_pi(n)) for n in (2..110)] # G. C. Greubel, May 17 2019

a(n) = A047783(n) - A000720(n).

cp's OEIS Frontend

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

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