A220475 - OEIS

%I #37 Nov 08 2014 21:47:30

%S 307,347,563,569,733,821,1427,1429,1433,1439,1447,1481,1867,1931,1973,

%T 2657,2659,2663,2671,2683,3187,3191,3313,3319,3323,3461,3511,3517,

%U 4001,4217,4231,4597,4621,4783,5387,5393,5413,5417,5477,5501,5639,5641,5651,6067,6311,6823,6857,7477,7523,7537

%N 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.

%C All terms are primes.

%C 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.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://arxiv.org/abs/1108.0475">Generalized Ramanujan primes</a>, arXiv 2011.

%H N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, <a href="http://link.springer.com/chapter/10.1007/978-1-4939-1601-6_1">Generalized Ramanujan primes</a>, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13

%H V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4

%H Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>

%F a(n) <= prime(32*(n+1)).

%t 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 *)

%t (* 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 *)

%Y Cf. A220293, A220462, A220463, A220474.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, _Charles R Greathouse IV_ and _Peter J. C. Moses_, Dec 15 2012

%E More terms from _Jean-François Alcover_, Feb 11 2013