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A220475
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Chebyshev numbers C_v(n) for v=15/14: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(14*x/15)>=n*log(x), where theta(x)=sum_{prime p<=x} log p.
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0
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307, 347, 563, 569, 733, 821, 1427, 1429, 1433, 1439, 1447, 1481, 1867, 1931, 1973, 2657, 2659, 2663, 2671, 2683, 3187, 3191, 3313, 3319, 3323, 3461, 3511, 3517, 4001, 4217, 4231, 4597, 4621, 4783, 5387, 5393, 5413, 5417, 5477, 5501, 5639, 5641, 5651, 6067, 6311, 6823, 6857, 7477, 7523, 7537
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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(100)=15013, all terms are (15/14)-Ramanujan numbers as in Shevelev's link, except for 821; the sequence is missing (15/4)-Ramanujan numbers 127 and 1423 and no others up to 15013.
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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(32*(n+1)).
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MATHEMATICA
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k=14; xs=Table[{m, Ceiling[x/.FindRoot[(x (-1300+Log[x]^4))/Log[x]^5==(k+1) m, {x, f[(k+1) m]-1}, AccuracyGoal->Infinity, PrecisionGoal->20, WorkingPrecision->100]]}, {m, 1, 101}]; Table[{m, 1+NestWhile[#-1&, xs[[m]][[2]], (1/Log[#1]Plus@@Log[Select[Range[Floor[(k #1)/(k+1)]+1, #1], PrimeQ]]&)[#]>m&]}, {m, 1, 100}] (* Peter J. C. Moses, Dec 20 2012 *)
(* Assuming range of x is from a(n) to 2*a(n) *) Clear[a, theta]; theta[x_] := theta[x] = Sum[Log[p], {p, Table[Prime[k], {k, 1, PrimePi[x]}]}] // N; a[0] = 293(* just to speed-up computation *); a[6] = 821(* the exception noted in comments *); a[n_] := a[ n] = (t = Table[an = Prime[pi]; Table[{an, x >= an && theta[x] - theta[14*x/15] >= n*Log[x]}, {x, an, 2*an}], {pi, PrimePi[a[n - 1]], 32*(n+1)}]; 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, 50}] (* Jean-François Alcover, Feb 11 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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EXTENSIONS
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STATUS
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approved
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