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A176614
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Conjectured least number M such that PrimePi(m)*log(m)/m lies between 1+(1/n) and 1-(1/n) for all m >= M.
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0
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1, 3, 3, 114, 294, 1674, 5904, 18596, 60181, 159875, 452639, 1297696, 3515851, 9840788, 27100113, 74045962, 198875487, 541400081
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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PROG
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(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)
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CROSSREFS
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KEYWORD
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more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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