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

1,28

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, ...

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

Formula

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

Example

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.

Mathematica

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

Crossrefs

Cf. A000720, A285586.

Sequence in context: A016357 A016388 A016004 * A359249 A284996 A215029

Adjacent sequences: A327799 A327800 A327801 * A327803 A327804 A327805

Keyword

nonn

Author

Bernard Schott, Sep 25 2019

Status

approved