A282870 -id:A282870 - 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

A359145 a(n) = smallest k such that li(k) - pi(k) >= n, where li(k) is the logarithmic integral and pi(x) is the number of primes <= x.

Original entry on oeis.org

6, 10, 27, 57, 95, 148, 221, 345, 539, 806, 1270, 1393, 1407, 1422, 2590, 2645, 3292, 4888, 4930, 5374, 7406, 7442, 8511, 11578, 11653, 11671, 11765, 11774, 18997, 19066, 19135, 19204, 19362, 19372, 30621, 31925, 31935, 31946, 31956, 47038, 47264, 55573, 64993

Offset: 1

N. J. A. Sloane, Feb 06 2023

nonn

Suggested by the "Great Prime Number Race", which investigates when li(n) - pi(n) changes sign.
Note this is different from the smallest k such that A052435(k) >= n, because of the rounding in A052435.
Since, by the prime number theorem li(n)/pi(n) converges to 1, this sequence is probably finite.

Roger Plymen, The Great Prime Number Race, AMS, 2020.

Patrick Demichel, The prime counting function and related subjects, April 05, 2005, 75 pages.

Cf. A000720, A052435, A282870.

seq[len_, kmax_] := Module[{s = Table[0, {len}], c = 0, k = 1, d}, While[c < len && k <= kmax, d = Floor[LogIntegral[k] - PrimePi[k]]; If[d > 0 && d <= len && s[[d]] == 0, Do[If[s[[i]] == 0, s[[i]] = k; c++], {i, 1, d}]]; k++]; s]; seq[50, 10^6] (* Amiram Eldar, Feb 07 2023 *)

More terms from Amiram Eldar, Feb 07 2023

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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