cp's OEIS Frontend

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.

%I A176614 #19 Feb 16 2025 08:33:12
%S A176614 1,3,3,114,294,1674,5904,18596,60181,159875,452639,1297696,3515851,
%T A176614 9840788,27100113,74045962,198875487,541400081
%N 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.
%C A176614 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.
%C A176614 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).
%C A176614 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
%H A176614 Pierre Dusart, <a href="https://arxiv.org/abs/1002.0442">Estimates of Some Functions Over Primes without R.H.</a>, arXiv:1002.0442 [math.NT], 2010.
%H A176614 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function</a>
%o A176614 (PARI) /* generates a(12) */
%o A176614 nmax=0;
%o A176614 for(n=1,2000000,v=primepi(n)*log(n)/n;if(v<1-(1/12)||v>1+(1/12),nmax=n));
%o A176614 print(nmax+1)
%Y A176614 Cf. A000720, A038625, A006880.
%K A176614 more,nonn
%O A176614 1,2
%A A176614 _Michael B. Porter_, Apr 21 2010
%E A176614 a(13)-a(18) from _Chai Wah Wu_, Apr 27 2018