A327802 Number of primes p such that n < p < (9/8) * n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3

Offset: 1

Author

Bernard Schott, Sep 25 2019

Keywords

• nonn

Comments

In 1932, Robert Hermann Breusch proved that for n > 47 there is at least one prime p between n and (9/8)*n. This was an improvement of Bertrand's postulate also called Chebyshev's theorem: if n > 1, there is always at least one prime p such that n < p < 2*n.

a(n) = 0 for 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 19, 20, 23, 24, 25, 31, 32, 47; the terms of A285586 correspond to the inequality n <= p <= (9/8) * n.

Records for a(n) = 0, 1, 2, 3, 4, ... are obtained for n = 1, 10, 28, 65, 96, ...

Examples

			9/8 * 17 = 19.125 and between 17 and 19.125, only 19 is a prime hence a(17) = 1.
9/8 * 39 = 43.875, and between 39 and 43.875, there are 41 and 43 that are primes hence a(39) = 2.
		

References

• François Le Lionnais, Jean Brette, Les Nombres remarquables, Hermann, 1983, nombre 48, page 46.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 48, page 106.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Robert Breusch, Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischen x und 2x stets Primzahlen liegen, Mathematische Zeitschrift (in German), December 1932, Volume 34, Issue 1, pp 505-526
• Wikipedia, Robert Breusch

Crossrefs

Cf. A000720, A285586.

Programs

Mathematica

Table[PrimePi[(9/8)*n] - PrimePi[n], {n, 1, 80}] (* Metin Sariyar, Sep 26 2019 *)

Formula

a(n) = pi(ceiling(9*n/8)-1) - pi(n), pi = A000720. - Alois P. Heinz, Sep 25 2019