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

1,1

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

Table of n, a(n) for n=1..50.

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

Cf. A220293, A220462, A220463.

Sequence in context: A153424 A100607 A092623 * A243767 A345533 A345785

Adjacent sequences: A220471 A220472 A220473 * A220475 A220476 A220477

Keyword

nonn

Author

Vladimir Shevelev, Charles R Greathouse IV and Peter J. C. Moses, Dec 15 2012

Extensions

More terms from Jean-François Alcover, Feb 11 2013

Status

approved