

A157755


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


0



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
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OFFSET

1,1


COMMENTS

From table in Kontorovich, version 2, 2005, page 7. Abstract: "The prime number theorem gives the following asymptotic for the nth 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 Vaughantype 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 PseudoTwin Primes Theorem, in "Multiple Dirichlet Series, Lfunctions and Automorphic Forms", Birkhauser Progress in Math Series, Vol. 300, (2012), 287298.


LINKS

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)


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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



