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

%I A327802 #38 May 21 2022 14:49:18
%S A327802 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,
%T A327802 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,
%U A327802 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
%N A327802 Number of primes p such that n < p < (9/8) * n.
%C A327802 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.
%C A327802 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.
%C A327802 Records for a(n) = 0, 1, 2, 3, 4, ... are obtained for n = 1, 10, 28, 65, 96, ...
%D A327802 François Le Lionnais, Jean Brette, Les Nombres remarquables, Hermann, 1983, nombre 48, page 46.
%D A327802 David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 48, page 106.
%H A327802 Alois P. Heinz, <a href="/A327802/b327802.txt">Table of n, a(n) for n = 1..20000</a>
%H A327802 Robert Breusch, <a href="https://doi.org/10.1007/BF01180606">Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischen x und 2x stets Primzahlen liegen</a>, Mathematische Zeitschrift (in German), December 1932, Volume 34, Issue 1, pp 505-526
%H A327802 Wikipedia, <a href="https://en.wikipedia.org/wiki/Robert_Breusch">Robert Breusch</a>
%F A327802 a(n) = pi(ceiling(9*n/8)-1) - pi(n), pi = A000720. - _Alois P. Heinz_, Sep 25 2019
%e A327802 9/8 * 17 = 19.125 and between 17 and 19.125, only 19 is a prime hence a(17) = 1.
%e A327802 9/8 * 39 = 43.875, and between 39 and 43.875, there are 41 and 43 that are primes hence a(39) = 2.
%t A327802 Table[PrimePi[(9/8)*n] - PrimePi[n], {n, 1, 80}] (* _Metin Sariyar_, Sep 26 2019 *)
%Y A327802 Cf. A000720, A285586.
%K A327802 nonn
%O A327802 1,28
%A A327802 _Bernard Schott_, Sep 25 2019