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%I #25 Apr 29 2023 14:08:14
%S 1,0,8,2,0,5,1,4,4,5,1,9,2,3,9,5,0,6,5,0,3,3,6,8,1,5,2,8,8,9,7,8,9,8,
%T 5,5,7,5,3,9,3,0,6,3,8,4,7,0,0,0,3,1,9,5,4,7,3,7,3,4,0,1,3,4,6,8,2,3,
%U 7,0,0,2,4,3,9,8,8,9,2,4,8,7,1
%N Decimal expansion of constant C = maximum value of 2*(Sum_{j=1..n} prime(j))/(n^2*log(n)).
%C According to Bach and Shallit (1996), Sum_{j=1..n} prime(j) ~ n^2*log(n)/2. Consequently, the function 2*(Sum_{j=1..n} prime(j))/(n^2*log(n)) tends to 1 as n tends to infinity; however, it has a maximum value of 1.0820514... when n=341276. In precise terms this constant is 802812394173*2/(341276^2*log(341276)) and it provides an upper bound for Sum_{j=1..n} prime(j) <= (802812394173*2/(341276^2*log(341276)))*n^2*log(n)/2 for all n >= 15. The prime sums tables of _R. J. Mathar_, A007504 show that a maximum for C occurs between n=200000 and n=400000. Further refinement gives the maximum value of C at n=341276 where the sum of primes from 2 through to 4889407, inclusively, gives 802812394173.
%D E. Bach and J. Shallit, Section 2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms, Cambridge, MIT Press, 1996.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>.
%F C = 802812394173*2/(341276^2*log(341276)).
%e 1.0820514451923950650336815288978985575393063847...
%t table=Table[2Sum[Prime[i], {i, 1, n}]/(n^2Log[n]), {n, 341200, 341400}]; max=Max[table]; n=1; While[table[[n]]!=max, n++]; Print[N[max, 100]," at n = ",n-1+341200]
%o (PARI) 1605624788346/(341276^2*log(341276)) \\ _Charles R Greathouse IV_, Apr 16 2012
%Y Cf. A007504.
%K nonn,cons
%O 1,3
%A _Frank M Jackson_, Apr 16 2012
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