A285586 -id:A285586 - cp's OEIS frontend

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

Bernard Schott, Sep 25 2019

nonn

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, ...

			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.

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.

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

Cf. A000720, A285586.

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

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

A331125 Numbers k such that there is no prime p between k and (9/8)k, exclusive.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 19, 20, 23, 24, 25, 31, 32, 47

Offset: 1

Bernard Schott, Jan 10 2020

In 1932, Robert Hermann Breusch proved that for n >= 48, there is at least one prime p between n and (9/8)n, exclusive (A327802).

The terms of A285586 correspond to numbers k such that there is no prime p between k and (9/8)n, inclusive.

			Between 16 and (9/8) * 16 = 18, exclusive, there is the prime 17, hence 16 is not a term.
Between 47 and (9/8) * 47 = 52.875, exclusive, 48, 49, 50, 51 and 52 are all composite numbers, hence 47 is a term.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised edition), Penguin Books, 1997, entry 48, p. 106.

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

Cf. A285586, A327802.

Select[Range[47], Count[Range[# + 1, 9# / 8], ?PrimeQ] == 0 &] (* _Amiram Eldar, Jan 11 2020 *)
Select[Range[1000], PrimePi[#] == PrimePi[9#/8] &] (* Alonso del Arte, Jan 16 2020 *)

A327802(a(n)) = 0.

A333241 Numbers k such that the number of primes p with k < p < (9/8) * k increases to a new record.

Original entry on oeis.org

1, 10, 28, 65, 96, 161, 177, 250, 341, 346, 412, 416, 540, 551, 586, 737, 785, 906, 924, 935, 976, 1004, 1159, 1162, 1180, 1386, 1393, 1397, 1408, 1441, 1840, 1852, 1856, 1857, 2055, 2119, 2124, 2128, 2193, 2199, 2202, 2490, 2492, 2519, 2528

Offset: 1

Bernard Schott, Mar 12 2020

nonn

In 1932, Robert Hermann Breusch proved that for n >= 48 there is at least one prime between n and (9/8)*n exclusive. This was an improvement of Bertrand's postulate, also called Chebyshev's theorem: if n > 1, there is always at least one prime between n and 2*n exclusive (A060715).

a(n) = k means that k is the first occurrence for which there are exactly n-1 primes p between k and (9/8)*k exclusive.

			a(6) = 161 since 163, 167, 173, 179, 181 are strictly between 161 and (9/8)*161 = 181.125 and it is the first time that 5 primes lie in an interval of this type.

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.

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. See also alternate link.
Wikipedia, Robert Breusch

Cf. A060715, A060756 (similar for Bertrand's postulate).
Cf. A014085 (Legendre's conjecture).
Cf. A285586, A327802, A331125.

f[n_] := PrimePi[9n/8] - PrimePi[n]; seq = {}; fmax = -1; Do[f1 = f[n]; If[f1 > fmax, fmax = f1; AppendTo[seq, n]], {n, 1, 2600}]; seq (* Amiram Eldar, Mar 12 2020 *)

f(n) = primepi(ceil(9*n/8) - 1) - primepi(n); \\ A327802
lista(nn) = {my(m=-1, nm, list = List()); for (n=1, nn, if ((nm=f(n)) > m, m = nm; listput(list, n));); Vec(list);} \\ Michel Marcus, Mar 23 2020

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

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A331125 Numbers k such that there is no prime p between k and (9/8)k, exclusive.

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A333241 Numbers k such that the number of primes p with k < p < (9/8) * k increases to a new record.

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