A217864 - OEIS

A217864

Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).

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

1,2

COMMENTS

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

REFERENCES

A. Brauer and H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.

LINKS

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.

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

An alternate version of A166712.

Cf. A217564, A096509, A000905, A050504, A000720.

Sequence in context: A028930 A112792 A138319 * A002100 A108352 A346149

Adjacent sequences: A217861 A217862 A217863 * A217865 A217866 A217867

KEYWORD

nonn

AUTHOR

Jon Perry, Oct 13 2012

STATUS

approved