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A333241
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Numbers k such that the number of primes p with k < p < (9/8) * k increases to a new record.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A014085 (Legendre's conjecture).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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