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

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

Author

David S. Newman, Feb 23 2017

Keywords

• sign

Comments

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.

References

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

Crossrefs

Cf. A000720, A047783, A052435, A359145.

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Sage

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

Formula

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