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%I #26 Mar 21 2021 00:06:17
%S 2,3,5,11,17,29,37,41,59,67,97,101,127,149,223,227,347,349,419,431,
%T 557,563,569,587,593,599,809,821,1423,1427,1429,1433,1447,1451,1481,
%U 1973,1987,1993,2657,2659,3299,3301,3307,3449,3457,3461,3511,3527,3529,5381,5387
%N Primes p such that theta(p)/p < theta(q)/q for all primes q > p, where theta(x) = log(x#)/x is Chebyshev's theta function and x# is the product of primes <= x, A034386(x).
%C 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).
%H Charles R Greathouse IV, <a href="/A215013/b215013.txt">Table of n, a(n) for n = 1..1000</a>
%H Pierre Dusart, <a href="http://arxiv.org/abs/1002.0442/">Estimates of some functions over primes without R.H.</a>, arXiv:1002.0442 [math.NT], 2010.
%H G. H. Hardy and J. E. Littlewood, <a href="http://dx.doi.org/10.1007/BF02422942">Contributions to the theory of the Riemann Zeta-Function and the theory of the distribution of primes</a>, Acta Mathematica 41 (1916), pp. 119-196.
%e 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.
%e Using Dusart's explicit bounds it suffices to check up to 2569516503 to verify the first thousand terms of this sequence.
%t 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 *)
%Y Cf. A108310.
%K nonn,nice
%O 1,1
%A _Charles R Greathouse IV_, Jul 31 2012