1,1

Every entry must be a prime.

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.)

When n=3745619057, 0.99999312926590387432389345880435140945170798255514 < log(n#)/n < 1. - Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 01 2005

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 (noe(AT)sspectra.com), Apr 17 2006

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 (raab-martin(AT)gmx.de), May 13 2008

J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x), Math. Comp. 29 (1975), no. 129, 243-269.

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).

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.

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 (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 01 2005)

Cf. A034386.

Sequence in context: A138184 A008965 A113864 this_sequence A104189 A119717 A000627

Adjacent sequences: A108307 A108308 A108309 this_sequence A108311 A108312 A108313

easy,nonn

Dave Rusin (rusin(AT)math.niu.edu), Jun 29 2005

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 01 2005