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A217864
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Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).
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0
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0, 2, 2, 2, 0, 2, 1, 2, 2, 1, 1, 2, 0, 1, 2, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n) is unbounded.
If Riemann Hypothesis is true, this is probably true as the PNT is generally a lower bound for Pi(n).
Conjecture: a(n)=0 infinitely often.
The first conjecture follows from Dickson's conjecture. The second conjecture follows from a theorem of Brauer & Zeitz on prime gaps. - Charles R Greathouse IV, Oct 15 2012
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REFERENCES
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A. Brauer and H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.
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LINKS
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Table of n, a(n) for n=1..87.
Paul Erdős, On the difference of consecutive primes, Quart. J. Math., Oxford Ser. 6 (1935), pp. 124-128.
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EXAMPLE
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log(1)=0 and 2*log(2) ~ 1.38629436112. Hence, a(1)=0.
Floor(2*log(2)) = 1 and 3*log(3) ~ 3.295836866. Hence, a(2)=2.
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MATHEMATICA
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Table[s = Floor[n*Log[n]]; PrimePi[(n+1) Log[n+1]] - PrimePi[s] + Boole[PrimeQ[s]], {n, 100}] (* T. D. Noe, Oct 15 2012 *)
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PROG
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(JavaScript)
function isprime(i) {
if (i==1) return false;
if (i==2) return true;
if (i%2==0) return false;
for (j=3; j<=Math.floor(Math.sqrt(i)); j+=2)
if (i%j==0) return false;
return true;
}
for (i=1; i<88; i++) {
c=0;
for (k=Math.floor(i*Math.log(i)); k<=(i+1)*Math.log(i+1); k++) if (isprime(k)) c++;
document.write(c+", ");
}
(PARI) a(n)=sum(k=n*log(n)\1, (n+1)*log(n+1), isprime(k)) \\ Charles R Greathouse IV, Oct 15 2012
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CROSSREFS
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An alternate version of A166712.
Cf. A217564, A096509, A000905, A050504, A000720.
Sequence in context: A028930 A112792 A138319 * A002100 A108352 A215883
Adjacent sequences: A217861 A217862 A217863 * A217865 A217866 A217867
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KEYWORD
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nonn
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AUTHOR
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Jon Perry, Oct 13 2012
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STATUS
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approved
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