OFFSET
1,1
COMMENTS
Sequence is infinite. This follows from the Prime Number Theorem in the form theta(x) ~ x, together with (5.111) in Hardy & Littlewood (p. 184).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Pierre Dusart, Estimates of some functions over primes without R.H., arXiv:1002.0442 [math.NT], 2010.
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann Zeta-Function and the theory of the distribution of primes, Acta Mathematica 41 (1916), pp. 119-196.
EXAMPLE
theta(5)/5 = log(2*3*5)/5 = 0.680 is less than theta(7)/7 = log(2*3*5*7)/7 = 0.763..., theta(11)/11 = 0.704..., etc.
Using Dusart's explicit bounds it suffices to check up to 2569516503 to verify the first thousand terms of this sequence.
MATHEMATICA
nMax = 1000; terms = 51; prd = 1; t = Table[p = Prime[n]; prd = prd*p; Log[prd]/p, {n, nMax}]; frst = 1; Table[mn = Min[Take[t, {frst, Length[t]}]]; pos = Position[t, mn, 1, 1][[1, 1]]; frst = pos + 1; Prime[pos], {terms}] (* T. D. Noe, Aug 02 2012 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Charles R Greathouse IV, Jul 31 2012
STATUS
approved