login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 10 11:50 EST 2016. Contains 279001 sequences.