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A209883 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. 0
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

Rosser, J. Barkley; Schoenfeld, Lowell Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 1962 64-94

LINKS

Table of n, a(n) for n=1..102.

Cornell University Library, Rosser, J. Barkley; Schoenfeld, Lowell Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 1962 64-94.

Mathworld, Prime Counting Function.

FORMULA

C = 30*log(113)/113 = 1.255058712932479796968707476181244691689202758...

EXAMPLE

The maximum value for PrimePi(n)*log(n)/n occurs at n = 113.

MATHEMATICA

$MaxPiecewiseCases=10000; sol=Maximize[{PrimePi[n]Log[n]/n, 1<n<10000}, n]; {N[sol[[1]], 100], sol[[2]]}

CROSSREFS

Cf. A000720, A057835.

Sequence in context: A061083 A177039 A075102 * A021396 A201317 A160177

Adjacent sequences:  A209880 A209881 A209882 * A209884 A209885 A209886

KEYWORD

nonn,cons

AUTHOR

Frank M Jackson, Mar 14 2012

STATUS

approved

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Last modified December 20 00:41 EST 2014. Contains 252240 sequences.