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A334599 a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n).

%I A334599 #30 Dec 15 2021 11:04:44
%S A334599 2,2,346,66942,7087878,744600720,85281842598,10892966758462,
%T A334599 1553240096780862,246080334487930558,43047454015229292840,
%U A334599 8262178422446205100776
%N A334599 a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n).
%C A334599 For a nonnegative integer m, pi(m) = A000720(m). It is well-known that if
%C A334599   m >= 17, then m/log(m) < pi(m). [Rosser and Schoenfeld]
%C A334599 Fix a real exponent d > 0. If m is big enough, then m < (m/log(m)) + (m/log(m))^(1 + d). In particular, choosing d = 1/n, with n >= 1, we deduce that a(n) exists.
%C A334599 Note that different choices of the exponent d will produce analogous sequences.
%C A334599 The estimates of pi(m) in [Dusart, Thm. 5.1] and [Axler, Thm. 2] allow us to obtain upper and lower bounds for a(n). In particular, we can conclude that in base 10:
%C A334599   a(13) has 25 digits, starting with 1729;
%C A334599   a(14) has 27 digits, starting with 392;
%C A334599   a(15) has 29 digits, starting with 962;
%C A334599   a(16) has 32 digits, starting with 2534.
%C A334599 The tool primecount [Walisch], used to compute pi(10^28) in A006880, can handle pi(m) for m <= 10^31, and since (a(n)) is monotonically increasing, it seems that the computation of a(n) for n >= 16 will be challenging.
%C A334599 It is easy to see that for every n >= 1, a(n) is even and a(n)+1 is prime. - _Eduard Roure Perdices_, Nov 07 2021
%H A334599 Christian Axler, <a href="http://math.colgate.edu/~integers/s61/s61.Abstract.html">Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality</a>, Integers 18 (2018), Paper No. A61, 14 pp.
%H A334599 Pierre Dusart, <a href="https://doi.org/10.1007/s11139-016-9839-4">Explicit estimates of some functions over primes</a>, The Ramanujan Journal 45 (2018), no. 1, 227-251.
%H A334599 J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Illinois J. Math. 6 (1962), no. 1, 64-94.
%H A334599 Kim Walisch, <a href="https://github.com/kimwalisch/primecount">primecount</a>, Github, Aug 14 2021.
%Y A334599 Cf. A334598, A087235, A038625.
%K A334599 nonn,more,hard
%O A334599 1,1
%A A334599 _Eduard Roure Perdices_, May 07 2020
%E A334599 a(8) from _Giovanni Resta_, May 07 2020
%E A334599 a(9)-a(10) from _Daniel Suteu_, May 20 2020
%E A334599 a(11)-a(12) from _Eduard Roure Perdices_, Nov 07 2021