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.

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

Author

N. J. A. Sloane, Feb 06 2023

Keywords

• nonn

Comments

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.

References

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

Links

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

Crossrefs

Cf. A000720, A052435, A282870.

Programs

Mathematica

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 *)

Extensions

More terms from Amiram Eldar, Feb 07 2023