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A182170
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Decimal expansion of constant C = maximum value of 2*(Sum_{j=1..n} prime(j))/(n^2*log(n)).
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0
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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, 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, 7, 0, 0, 2, 4, 3, 9, 8, 8, 9, 2, 4, 8, 7, 1
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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E. Bach and J. Shallit, Section 2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms, Cambridge, MIT Press, 1996.
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LINKS
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Eric Weisstein's World of Mathematics, Prime Sums.
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FORMULA
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C = 802812394173*2/(341276^2*log(341276)).
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EXAMPLE
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1.0820514451923950650336815288978985575393063847...
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MATHEMATICA
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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]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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