A111499 a(n) = floor(10^n/PrimePi(10^n)) - 1.

1, 3, 4, 7, 9, 11, 14, 16, 18, 20, 23, 25, 27, 30, 32, 34, 37, 39, 41, 44, 46, 48, 50, 53, 55, 57, 60, 62, 64, 67, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 92, 94, 97, 99, 101, 103, 106, 108, 110, 113, 115, 117, 120, 122, 124, 126, 129, 131, 133, 136, 138, 140, 143

Offset

1,2

Comments

PrimePi(n) is the number of primes less than or equal to n.

10^n/PrimePi(10^n) - 1) is the ratio of the number of composite numbers less than 10^n divided by the number of prime numbers less than 10^n. Conjecture: Except for the first 2 terms, the difference between successive terms is 2 or 3.

Many terms can be obtained via the following bounds by Pierre Dusart: (1 + 1/log(x)^2 + 2/log(x)^2) * x/log(x) < primepi(x) < (1 + 1/log(x)^2 + 2/log(x)^2 + 7.59/log(x)^3) * x/log(x), for x >= 88789. - Giovanni Resta, Jan 03 2020

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal (2018), 45: 227-251.

XavierGourdon, Numbers, Constants and computations.

Formula

a(n) = floor(10^n / A006880(n)) - 1. - Andrew Howroyd, Jan 02 2020

Mathematica

f[n_] := Floor[10^n/PrimePi[10^n] - 1]; Table[ f[n], {n, 14}] (* Robert G. Wilson v, Nov 18 2005 *)
piB[x_] := If[x < 10^5, PrimePi[x] {1, 1}, x/Log[x] (1 + 1/Log[x] + 2/Log[x]^2 + {0, 7.59}/Log[x]^3)]; f[n_] := Floor[10^n / piB[10^n]] - 1; Reap[ Do[ If [Length[u = Union@ f@ n] > 1, Break[], Sow@ u[[1]]], {n, 1000}]][[2, 1]] (* Giovanni Resta, Jan 03 2020 *)

Prog

(PARI) PiRatio(m, n) = /* Good only up to 10^9 */ { local(x, p1, p2, a, b); for(x=m, n, p1=10^x; a=floor(p1/primepi(p1)-1); print1(a, ", ") ) }

Crossrefs

Cf. A000720 (PrimePi), A006880 (number of primes < 10^n).

Sequence in context: A245239 A191978 A187477 * A077025 A280493 A032729

Adjacent sequences: A111496 A111497 A111498 * A111500 A111501 A111502

Keyword

nonn

Author

Cino Hilliard, Nov 16 2005

Extensions

a(23) from Robert G. Wilson v, Nov 18 2005

a(24)-a(27) from Andrew Howroyd, Jan 02 2020

Terms a(28) and beyond from Giovanni Resta, Jan 03 2020

Status

approved