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A209883
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Decimal expansion of constant C = maximum value that PrimePi(n)*log(n)/n reaches where PrimePi(n) is the number of primes less than or equal to n, A000720.
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0
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1, 2, 5, 5, 0, 5, 8, 7, 1, 2, 9, 3, 2, 4, 7, 9, 7, 9, 6, 9, 6, 8, 7, 0, 7, 4, 7, 6, 1, 8, 1, 2, 4, 4, 6, 9, 1, 6, 8, 9, 2, 0, 2, 7, 5, 8, 0, 6, 2, 7, 4, 1, 7, 1, 5, 4, 1, 7, 7, 9, 1, 5, 1, 3, 8, 0, 8, 0, 2, 8, 4, 7, 0, 5, 0, 2, 4, 0, 2, 6, 7, 3, 6, 7, 3, 3, 2, 4, 8, 0, 5, 9, 7, 3, 4, 1, 7, 3, 6, 5, 8, 3
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OFFSET
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1,2
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COMMENTS
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The prime number theorem states that PrimePi(n) ~ n/log(n). Consequently, the function PrimePi(n)*log(n)/n tends to 1 as n tends to infinity, however it has a maximum value of 1.2550587.... when n=113. In precise terms this constant is 30*log(113)/113 and it provides an upper bound for PrimePi(n), i.e. PrimePi(n) <= (30*log(113)/113)*n/log(n) for all n>1.
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LINKS
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FORMULA
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C = 30*log(113)/113 = 1.255058712932479796968707476181244691689202758...
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EXAMPLE
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The maximum value for PrimePi(n)*log(n)/n occurs at n = 113.
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MATHEMATICA
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$MaxPiecewiseCases=10000; sol=Maximize[{PrimePi[n]Log[n]/n, 1<n<10000}, n]; {N[sol[[1]], 100], sol[[2]]}
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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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