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A220462
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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.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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
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FORMULA
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a(n)<=prime(4*(n+1)).
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MATHEMATICA
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(* 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-François Alcover, Jan 24 2013 *)
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CROSSREFS
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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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