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
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
KEYWORD
nonn
AUTHOR
Bernard Schott, Mar 12 2020
STATUS
approved