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%I A333241 #65 May 21 2022 14:49:34
%S A333241 1,10,28,65,96,161,177,250,341,346,412,416,540,551,586,737,785,906,
%T A333241 924,935,976,1004,1159,1162,1180,1386,1393,1397,1408,1441,1840,1852,
%U A333241 1856,1857,2055,2119,2124,2128,2193,2199,2202,2490,2492,2519,2528
%N A333241 Numbers k such that the number of primes p with k < p < (9/8) * k increases to a new record.
%C A333241 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).
%C A333241 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.
%D A333241 François Le Lionnais & Jean Brette, Les Nombres remarquables, Hermann, 1983, nombre 48, page 46.
%D A333241 David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 48, page 106.
%H A333241 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. See also <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002373270">alternate link</a>.
%H A333241 Wikipedia, <a href="https://en.wikipedia.org/wiki/Robert_Breusch">Robert Breusch</a>
%e A333241 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.
%t A333241 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 *)
%o A333241 (PARI) f(n) = primepi(ceil(9*n/8) - 1) - primepi(n); \\ A327802
%o A333241 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
%Y A333241 Cf. A060715, A060756 (similar for Bertrand's postulate).
%Y A333241 Cf. A014085 (Legendre's conjecture).
%Y A333241 Cf. A285586, A327802, A331125.
%K A333241 nonn
%O A333241 1,2
%A A333241 _Bernard Schott_, Mar 12 2020

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