A220463 Chebyshev numbers C_v(n) for v=1.2: a(n) is the smallest number such that if x>=a(n), then theta(x)-theta(5*x/6)>=n*log(x), where theta(x)=sum_{prime p<=x}log p.

59, 137, 139, 149, 223, 241, 347, 353, 383, 389, 563, 569, 593, 613, 631, 641, 821, 823, 853, 929, 937, 1009, 1013, 1061, 1069, 1277, 1279, 1361, 1427, 1433, 1481, 1487, 1597, 1601, 1607, 1609, 1613, 1973, 1979, 1997, 2011, 2081, 2083, 2113, 2203, 2269, 2273, 2297

Offset: 1

Author

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

Keywords

• nonn

Comments

All terms are primes.

Up to a(98)=5381, all terms are 1.2-Ramanujan numbers as in Shevelev's link; up to 5381, the only missing 1.2-Ramanujan numbers are 29 and 5171.

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

Crossrefs

Cf. A220293, 220462.

Programs

Mathematica

k=5; 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 *)

Formula

a(n)<=prime(11*(n+1)).