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

1,1

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

Table of n, a(n) for n=1..48.

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(11*(n+1)).

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 *)

Crossrefs

Cf. A220293, 220462.

Sequence in context: A139994 A107157 A039537 * A142171 A129480 A044310

Adjacent sequences: A220460 A220461 A220462 * A220464 A220465 A220466

Keyword

nonn

Author

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

Status

approved