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
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
KEYWORD
more,nonn
AUTHOR
Michael B. Porter, Apr 21 2010
EXTENSIONS
a(13)-a(18) from Chai Wah Wu, Apr 27 2018
STATUS
approved