

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 6494


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 6494.
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



