A108310 - OEIS

%I #22 Oct 17 2017 04:00:49

%S 2,3,5,7,13,19,43,47,73,103,107,109,113,199,283,467,661,887,1063,1069,

%T 1097,1103,1109,1123,1129,1303,1307,1321,1327,1621,1627,2803,3931,

%U 3947,4273,4289,4297,5867,5869,5881,6373,6379,9439,9473,9479,9497,9551,9859

%N Successive maxima of log(n#)/n where n# is the product of the primes less than n.

%C Every entry must be a prime.

%C Note that log(n#)=theta(n) (the Chebyshev function) for which bounds are known (e.g. Rosser and Schoenfeld have an estimate |theta(n)-n| < n/(40 log n).) In particular, log(n#)/n tends to 1, which allows a proof of the Prime Number Theorem. I suspect log(n#) can be greater than n for some n, which would make the sequence finite, but I do not know an example of such an n. (When n=30337841, 0.9999 < log(n#)/n < 1.)

%C When n=3745619057, 0.99999312926590387432389345880435140945170798255514 < log(n#)/n < 1. - _Robert G. Wilson v_, Jul 01 2005

%C Computational experiments show that it may be true that n > log(n#) for all n. In fact, it appears that, for any k, n > log(n#) + k*log(n) except for a finite number of small primes. For k=1, only 5, 7 and 19 are the exceptional n. This inequality is still consistent with 1 being the limiting value of log(n#)/n. - _T. D. Noe_, Apr 17 2006

%C Apparently in the long run (n-theta(n))/(Li(n)-Pi(n)) goes to log(n), so if Li(n)<Pi(n), which will happen before 1.4x10^316, then clearly there are values n for which n<theta(n). (According to my calculations, theta(n) will surpass n even a while before Pi(n) surpasses Li(n).) - _Martin Raab_, May 13 2008

%C Sequence is finite since psi(x) - x is greater than sqrt x * log log log x infinitely often, and hence theta(x) > x infinitely often [but theta(x) - x = o(x), see Rosser & Schoenfeld]. See Hardy & Littlewood section 5. - _Charles R Greathouse IV_, Aug 02 2012

%H Charles R Greathouse IV, <a href="/A108310/b108310.txt">Table of n, a(n) for n = 1..10000</a>

%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.

%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0457373-7">Sharper bounds for the Chebyshev functions theta(x) and psi(x)</a>, Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.

%e 13 follows 7 because log(7#)/7 = log(210)/7 = 0.7638, while log(8#)/8 and so on are smaller but log(13#)/13= 0.7931 is larger. A larger entry is 3445943 since log(n#)<0.99978 n for smaller n but log(3445943#)=3445185.8713457=(0.999780284)(3445943).

%p A:=[]:b:=0:S:=0:n:=1: while true do n:=nextprime(n): S:=S+evalf(log(n)): if S>b*n then A:=[op(A),n]: b:= S/n: fi: od: #Program must be terminated manually! Array "A" is the sequence.

%t lmt = slp = 0; lst = {}; Do[p = Prime[n]; slp = slp + N[Log[p], 12]; If[slp/p > lmt, lmt = slp/p; AppendTo[lst, p]], {n, 1224}]; lst (* _Robert G. Wilson v_, Jul 01 2005 *)

%o (PARI) r=th=0; forprime(p=2, 1e6, th+=log(p); t=th/p; if(t>r, r=t; print1(p", "))) \\ _Charles R Greathouse IV_, Dec 17 2014

%Y Cf. A034386, A215013.

%K nonn,fini

%O 1,1

%A _David J. Rusin_, Jun 29 2005

%E More terms from _Robert G. Wilson v_, Jul 01 2005