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

1,2

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.

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

Table of n, a(n) for n=1..45.

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Cf. A014085 (Legendre's conjecture).

Cf. A285586, A327802, A331125.

Sequence in context: A269441 A048491 A124703 * A251319 A126364 A076712

Adjacent sequences: A333238 A333239 A333240 * A333242 A333243 A333244

Keyword

nonn

Author

Bernard Schott, Mar 12 2020

Status

approved