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

2, 2, 346, 66942, 7087878, 744600720, 85281842598, 10892966758462, 1553240096780862, 246080334487930558, 43047454015229292840, 8262178422446205100776

Offset: 1

Author

Eduard Roure Perdices, May 07 2020

Keywords

• nonn
• more
• hard

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(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(10^28) 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.

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

Links

• 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, Aug 14 2021.

Crossrefs

Cf. A334598, A087235, A038625.

Extensions

a(8) from Giovanni Resta, May 07 2020

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

a(11)-a(12) from Eduard Roure Perdices, Nov 07 2021