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

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

Author

Bernard Schott, Mar 12 2020

Keywords

• nonn

Comments

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.

Examples

			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.
		

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

• 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

Crossrefs

Cf. A060715, A060756 (similar for Bertrand's postulate).

Cf. A014085 (Legendre's conjecture).

Cf. A285586, A327802, A331125.

Programs

Mathematica

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 *)

PARI

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