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A220462 Chebyshev numbers C_v(n) for v=3/2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(2*x/3)>=n*log(x), where theta(x) = sum_{prime p<=x} log p. 2
13, 37, 41, 67, 73, 97, 127, 137, 173, 179, 181, 211, 229, 239, 263, 307, 311, 347, 367, 379, 431, 433, 443, 449, 479, 487, 541, 563, 587, 599, 607, 641, 643, 673, 739, 757, 787, 797, 809, 823, 827, 859, 937, 967, 997, 1019, 1031, 1039, 1049, 1061, 1087 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

All terms are primes.

Up to a(97)=2333, only four terms of the sequence (a(33)=643, a(34)=673, a(76)=1721 and a(77)=1741) are not (3/2)-Ramanujan numbers as in Shevelev's link; up to 2333, the only (3/2)-Ramanujan numbers missing from the sequence are 2, 617, 653, 709, 1709, 1733, and 1747.

LINKS

Table of n, a(n) for n=1..51.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, arXiv 2011.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13

V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

FORMULA

a(n)<=prime(4*(n+1)).

MATHEMATICA

(* Assuming range of x is from a(n) to 2*a(n) *) theta[x_] := Sum[Log[p], {p, Table[Prime[k], {k, 1, PrimePi[x]}]}]; Clear[a]; a[0] = 2; a[n_] := a[n] = (t = Table[{an, x >= an && theta[x] - theta[2*(x/3)] >= n*Log[x]}, {an, a[n - 1], Prime[4*(n + 1)]}, {x, an, 2*an}]; sp = t // Flatten[#, 1] & // Sort // Split[#, #1[[1]] == #2[[1]] &] &; Select[sp, And @@ (#[[All, 2]]) &] // First // First // First); Table[Print[a[n]]; a[n], {n, 1, 51}] (* Jean-Fran├žois Alcover, Jan 24 2013 *)

CROSSREFS

Cf. A220293.

Sequence in context: A088963 A301591 A301857 * A280997 A185006 A285887

Adjacent sequences:  A220459 A220460 A220461 * A220463 A220464 A220465

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 15 2012

STATUS

approved

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)