A157755 Primes that hit the value suggested by the prime number theorem "on the nose" according to Kontorovich.

5, 7, 17, 23, 37, 113, 137, 157, 167, 173, 193, 199, 281, 373, 379, 397, 409, 421, 433, 577, 641, 647, 673, 719, 739, 839, 859, 941, 947, 1009, 1051, 1093, 1163, 1213, 1277, 1291, 1327, 1399, 1487, 1523, 1553, 1567, 1597, 1619, 1663, 1693, 1723, 1753, 1873, 1933, 1979

Offset

1,1

Comments

From table in Kontorovich, version 2, 2005, page 7. Abstract: "The prime number theorem gives the following asymptotic for the n-th prime: p_n ~ iL(n), where we are calling iL the inverse to the logarithmic integral function, Li. Let pi(x) denote the number of primes p <= x with p=[iL(n)] for some n. We say that these primes hit the value suggested by the prime number theorem "on the nose". Using exponential sums, the method of stationary phase, and Vaughan-type identities, we show that pi(x) ~ x/log^2(x) and interpret this fact as the independence of the process of the primes to their average value, iL."

For the benefit of anyone trying to track down the history of the sequence, we give references to all four versions of the Kontorovich paper (only one of which gives a list of the terms).

References

Alex V. Kontorovich, A Pseudo-Twin Primes Theorem, in "Multiple Dirichlet Series, L-functions and Automorphic Forms", Birkhauser Progress in Math Series, Vol. 300, (2012), 287--298.

Links

Table of n, a(n) for n=1..51.

Alex V. Kontorovich, The Prime Number Theorem on the Nose, arXiv:math/0507569v1.pdf, version 1, 2005, 5 pages.

Alex V. Kontorovich, The Prime Number Theorem on the Nose, arXiv:math/0507569v2.pdf, version 2, 2005, 11 pages. (This is the only version of the paper that gives the sequence explicitly)

Alex V. Kontorovich, A pseduo-twin primes theorem, arXiv:math/0507569v3.pdf, version 3, 2010, 11 pages.

Example

a(10) = because for k = 40, which is the 10th such value in the cited table, n = 44, p(k) = 173, iL(n) = 173.094, Li(p(k)) = 43.981, Li(p(k)+1) = 44.175, 1 / log(p(k)) = 0.194.

Mathematica

Li[x_]:=LogIntegral[x]-LogIntegral[2];
Select[Table[Floor@x/.FindRoot[Li[x]==k, {x, Prime@k}], {k, 3, 1000}], PrimeQ] (* Giorgos Kalogeropoulos, Sep 01 2023 *)

Crossrefs

Cf. A000040.

Sequence in context: A145354 A214345 A166109 * A265812 A128352 A025085

Adjacent sequences: A157752 A157753 A157754 * A157756 A157757 A157758

Keyword

nonn

Author

Jonathan Vos Post, Mar 05 2009

Extensions

Entry revised by N. J. A. Sloane, Sep 13 2014

More terms by Giorgos Kalogeropoulos, Sep 01 2023

Status

approved