|
| |
|
|
A217864
|
|
Number of prime numbers between floor(n*log(n)) and (n + 1)log(n + 1).
|
|
0
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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 A215883
Adjacent sequences: A217861 A217862 A217863 * A217865 A217866 A217867
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Jon Perry, Oct 13 2012
|
|
|
STATUS
|
approved
|
| |
|
|
