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A108310
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Successive maxima of log(n#)/n where n# is the product of the primes less than n.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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)<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
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
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LINKS
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EXAMPLE
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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).
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MAPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,fini
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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