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.

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

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(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

Cf. A220293, A220462, A220463, A220474.

Sequence in context: A128477 A028680 A118282 * A282594 A050201 A142376

Adjacent sequences: A220472 A220473 A220474 * A220476 A220477 A220478

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