

A176614


Conjectured least number M such that PrimePi(m)*log(m)/m lies between 1+(1/n) and 1(1/n) for all m >= M.


0



1, 3, 3, 114, 294, 1674, 5904, 18596, 60181, 159875, 452639, 1297696, 3515851, 9840788, 27100113, 74045962, 198875487, 541400081
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OFFSET

1,2


COMMENTS

The Prime Number Theorem guarantees that the terms of this sequence exist. It states that the limit of PrimePi(x)*log(x)/x as x goes to infinity exists and is equal to 1. The existence of M follows from the definition of a limit. The terms a(n) listed are taken from a table of PrimePi(m)*log(m)/m up to m=2*10^6.
The function PrimePi(m)*log(m)/m is less than 1.25506 for all m > 1 and is greater than 1 for all m >= 17, so the first three a(n) are proved (see the Weisstein reference).
Using the bounds given by Dusart (2010), each of the terms a(1)a(18) has been verified to be the minimal number M for n = 1..18.  Chai Wah Wu, Apr 27 2018


LINKS

Table of n, a(n) for n=1..18.
Pierre Dusart, Estimates of Some Functions Over Primes without R.H., arXiv:1002.0442 [math.NT], 2010.
Eric W. Weisstein, Prime Counting Function


PROG

(PARI) /* generates a(12) */
nmax=0;
for(n=1, 2000000, v=primepi(n)*log(n)/n; if(v<1(1/12)v>1+(1/12), nmax=n));
print(nmax+1)


CROSSREFS

Cf. A000720, A038625, A006880.
Sequence in context: A325486 A206485 A009491 * A173797 A173127 A230646
Adjacent sequences: A176611 A176612 A176613 * A176615 A176616 A176617


KEYWORD

more,nonn


AUTHOR

Michael B. Porter, Apr 21 2010


EXTENSIONS

a(13)a(18) from Chai Wah Wu, Apr 27 2018


STATUS

approved



