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A220475
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.
0
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
OFFSET
1,1
COMMENTS
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.
LINKS
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(32*(n+1)).
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
EXTENSIONS
More terms from Jean-François Alcover, Feb 11 2013
STATUS
approved