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.

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

Offset: 1

Author

Michael B. Porter, Apr 21 2010

Keywords

• more
• nonn

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

Crossrefs

Cf. A000720, A038625, A006880.

Programs

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)

Extensions

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