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

Offset: 1

Author

Jon Perry, Oct 13 2012

Keywords

• nonn

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

Examples

			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.
		

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

• Paul Erdős, On the difference of consecutive primes, Quart. J. Math., Oxford Ser. 6 (1935), pp. 124-128.

Crossrefs

An alternate version of A166712.

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

Programs

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+", ");
}

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 *)

PARI

a(n)=sum(k=n*log(n)\1,(n+1)*log(n+1),isprime(k)) \\ Charles R Greathouse IV, Oct 15 2012