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.
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
Keywords
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
Programs
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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 *)
Formula
a(n)<=prime(31*(n+1)).
Extensions
More terms from Jean-François Alcover, Feb 11 2013
Comments