A108310 Successive maxima of log(n#)/n where n# is the product of the primes less than n.

2, 3, 5, 7, 13, 19, 43, 47, 73, 103, 107, 109, 113, 199, 283, 467, 661, 887, 1063, 1069, 1097, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1621, 1627, 2803, 3931, 3947, 4273, 4289, 4297, 5867, 5869, 5881, 6373, 6379, 9439, 9473, 9479, 9497, 9551, 9859

Offset: 1

Author

David J. Rusin, Jun 29 2005

Keywords

• nonn
• fini

Comments

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, 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, Apr 17 2006

Apparently in the long run (n-theta(n))/(Li(n)-Pi(n)) goes to log(n), so if Li(n)Martin Raab, May 13 2008

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

Examples

			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).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• 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.
• J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x), Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.

Crossrefs

Cf. A034386, A215013.

Programs

Maple

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.

Mathematica

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 *)

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

Extensions

More terms from Robert G. Wilson v, Jul 01 2005