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A334599 a(n) is the largest nonnegative integer m such that m - pi(m) >= pi(m)^(1 + 1/n). 1
2, 2, 346, 66942, 7087878, 744600720, 85281842598, 10892966758462, 1553240096780862, 246080334487930558 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For a nonnegative integer m, pi(m) = A000720(m). It is well-known that if

  m >= 17, then m/log(m) < pi(m). [Rosser and Schoenfeld]

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.

Note that different choices of the exponent d will produce analogous sequences.

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:

  a(11) has 20 digits, starting with 430;

  a(12) has 22 digits, starting with 826;

  a(13) has 25 digits, starting with 1729;

  a(14) has 27 digits, starting with 392;

  a(15) has 29 digits, starting with 962;

  a(16) has 32 digits, starting with 2534.

The tool primecount [Walisch], used to compute pi(10^27) 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.

LINKS

Table of n, a(n) for n=1..10.

Christian Axler, Estimates for pi(x) for large values of x and Ramanujan's prime counting inequality, Integers 18 (2018), Paper No. A61, 14 pp.

Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal 45 (2018), no. 1, 227-251.

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64-94.

Kim Walisch, primecount, Github, Apr 07 2020.

CROSSREFS

Cf. A334598, A087235, A038625.

Sequence in context: A119512 A262058 A067091 * A013556 A093596 A095304

Adjacent sequences:  A334596 A334597 A334598 * A334600 A334601 A334602

KEYWORD

nonn,more,hard

AUTHOR

Eduard Roure Perdices, May 07 2020

EXTENSIONS

a(8) from Giovanni Resta, May 07 2020

a(9)-a(10) from Daniel Suteu, May 20 2020

STATUS

approved

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Last modified August 1 10:37 EDT 2021. Contains 346385 sequences. (Running on oeis4.)