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A220474
Chebyshev numbers C_v(n) for v=10/9: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(9*x/10)>=n*log(x), where theta(x)=sum_{prime p<=x}log p.
1
223, 227, 269, 349, 359, 569, 587, 593, 739, 809, 857, 991, 1009, 1019, 1259, 1481, 1483, 1487, 1489, 1861, 1867, 1993, 1997, 2003, 2027, 2267, 2269, 2657, 2671, 2687, 2689, 2699, 3181, 3187, 3307, 3313, 3319, 3323, 3457, 3461, 3491, 3527, 3529, 3581, 3623, 3769, 4049, 4201, 4391, 4481
OFFSET
1,1
COMMENTS
All terms are primes.
Up to a(99)=9029, all terms are (10/9)-Ramanujan numbers as in Shevelev's link; up to 9029, the only missing (10/9)-Ramanujan number is 127.
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(31*(n+1)).
MATHEMATICA
k=9; 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] = 211(* just to speed-up computation *); a[n_] := a[ n] = (t = Table[an = Prime[pi]; Table[{an, x >= an && theta[x] - theta[9*x/10] >= n*Log[x]}, {x, an, 2*an}], {pi, PrimePi[a[n-1]], 31*(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