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%I #23 Apr 27 2019 05:23:20
%S 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,
%T 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,
%U 1,0,1,3,2,0,0,1,1,0,2,1,1,0,1,1,2,1,1
%N Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).
%C Conjecture: a(n) is unbounded.
%C If Riemann Hypothesis is true, this is probably true as the PNT is generally a lower bound for Pi(n).
%C Conjecture: a(n)=0 infinitely often.
%C 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
%D A. Brauer and H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.
%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1935-07.pdf">On the difference of consecutive primes</a>, Quart. J. Math., Oxford Ser. 6 (1935), pp. 124-128.
%e log(1)=0 and 2*log(2) ~ 1.38629436112. Hence, a(1)=0.
%e Floor(2*log(2)) = 1 and 3*log(3) ~ 3.295836866. Hence, a(2)=2.
%t 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 *)
%o (JavaScript)
%o function isprime(i) {
%o if (i==1) return false;
%o if (i==2) return true;
%o if (i%2==0) return false;
%o for (j=3;j<=Math.floor(Math.sqrt(i));j+=2)
%o if (i%j==0) return false;
%o return true;
%o }
%o for (i=1;i<88;i++) {
%o c=0;
%o for (k=Math.floor(i*Math.log(i));k<=(i+1)*Math.log(i+1);k++) if (isprime(k)) c++;
%o document.write(c+", ");
%o }
%o (PARI) a(n)=sum(k=n*log(n)\1,(n+1)*log(n+1),isprime(k)) \\ _Charles R Greathouse IV_, Oct 15 2012
%Y An alternate version of A166712.
%Y Cf. A217564, A096509, A000905, A050504, A000720.
%K nonn
%O 1,2
%A _Jon Perry_, Oct 13 2012
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