A096509 -id:A096509 - cp's OEIS frontend

A096834 Singular primes mentioned in A096833 around the listed primorials.

211, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309

Offset: 1

Labos Elemer, Jul 14 2004

nonn

Neighborhoods of most primorials (=center) are either empty or poor of primes. The listed primes arise if center=A002110(A096833(n)), while radius=ceiling(log(center)).

The subsequent terms are too large to include: a(6) = A002110(66)-1, which has 131 digits. a(7) = A002110(68)-1, which has 136 digits. a(8) = A002110(75)+1, which has 154 digits. - David Wasserman, Nov 16 2007

			n=6: around 30030, the prime in question is 30029;
n=66: around A002110[66], the single prime has more than 121 decimal digits.

Cf. A096509-A096523, A096830-A096840; A002110.

A096835 Number of primes in the neighborhood of 3^n with radius ceiling(log(3^n)).

Original entry on oeis.org

3, 2, 3, 2, 2, 2, 1, 3, 2, 2, 0, 0, 1, 3, 3, 0, 2, 2, 2, 3, 2, 4, 1, 3, 3, 2, 3, 2, 1, 2, 2, 1, 0, 1, 2, 5, 2, 3, 0, 2, 4, 1, 0, 3, 3, 2, 2, 1, 3, 3, 2, 1, 2, 3, 2, 2, 5, 0, 3, 2, 2, 3, 4, 0, 1, 3, 0, 1, 4, 0, 2, 1, 1, 2, 3, 2, 3, 1, 2, 3, 3, 0, 0, 1, 2, 2, 2, 2, 2, 2, 5, 3, 0, 1, 6, 1, 4, 5, 1, 2, 3, 2, 1, 1, 2

Offset: 1

Labos Elemer, Jul 14 2004

nonn

Cf. A096509-A096523, A096830-A096840.

a(n) equals almost A096509(3^n) >= a(n) because here only primes are counted, the true prime powers not.

A096836 Number of primes in the neighborhood of 10^n with radius ceiling(log(10^n)).

Original entry on oeis.org

3, 3, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 1, 0, 7, 5, 1, 3, 0, 2, 2, 2, 1, 0, 0, 0, 2, 3, 3, 2, 4, 0, 2, 1, 3, 1, 5, 2, 0, 2, 4, 4, 2, 1, 3, 3, 3, 1, 0, 0, 0, 1, 2, 2, 1, 0, 1, 4, 4, 1, 2, 5, 2, 1, 4, 2, 3, 3, 2, 3, 3, 2, 1, 1, 2, 3, 1, 4, 4, 0, 4, 0, 2, 2, 0, 2, 3, 2, 6, 0, 3, 4, 4, 1, 1, 4, 0, 0, 4, 3, 3, 0, 3

Offset: 1

Labos Elemer, Jul 14 2004

nonn

			a(5)=3 and the 3 primes in [99988,100012] are {99989,99991,100003}.

Cf. A096509-A096523, A096830-A096840.

a(n) equals almost A096509(10^n) >= a(n) because here only primes are counted, the true prime powers not.

A096837 Number of primes in the neighborhood of n^n with radius ceiling(log(n^n)).

Original entry on oeis.org

0, 3, 3, 2, 2, 2, 3, 2, 2, 1, 0, 1, 3, 3, 1, 2, 1, 0, 6, 1, 2, 2, 3, 1, 1, 3, 3, 1, 0, 3, 2, 4, 2, 2, 4, 3, 2, 1, 5, 1, 1, 2, 3, 1, 3, 0, 2, 1, 3, 2, 1, 0, 4, 1, 2, 2, 2, 2, 1, 0, 1, 3, 2, 1, 0, 0, 2, 1, 3, 1, 1, 2, 1, 0, 2, 3, 5, 3, 3, 0, 3, 2, 2, 4, 4, 0, 6, 2, 1, 2, 1, 3, 3, 2, 1, 3, 2, 2, 1, 4, 1, 6, 0, 2, 1

Offset: 1

Labos Elemer, Jul 14 2004

nonn

			a(3)=3 and the 3 primes in [23,31] are {23,29,31}.

Cf. A096509-A096523, A096830-A096840.

a(n) almost equals A096509(n^n) >= a(n) because here only primes are counted, the true prime powers not.

A096838 Number of primes in the neighborhood of prime(n), the n-th prime in the center with radius ceiling(log(prime(n))).

Original entry on oeis.org

2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 4, 4, 3, 1, 3, 4, 4, 4, 3, 2, 2, 3, 3, 3, 3, 4, 3, 3, 1, 3, 4, 4, 3, 2, 2, 3, 3, 4, 2, 2, 3, 3, 3, 2, 2, 2, 1, 2, 2, 2, 3, 3, 3, 2, 3, 4, 4, 3, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3

Offset: 1

Labos Elemer, Jul 14 2004

nonn

			a(100)=2 and the two primes in [534,548] are {541,547};

Cf. A096509-A096523, A096830-A096840.

a(n) almost equals A096509(prime(n)) >= a(n) because here only primes are counted, the true prime powers not.

For all n, a(n) >= 1.

A217864 Number of prime numbers between floor(nlog(n)) and (n + 1)log(n + 1).

Original entry on oeis.org

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

Jon Perry, Oct 13 2012

nonn

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

			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.

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

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

An alternate version of A166712.

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

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+", ");
}

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 *)

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

cp's OEIS Frontend

A096834 Singular primes mentioned in A096833 around the listed primorials.

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A096835 Number of primes in the neighborhood of 3^n with radius ceiling(log(3^n)).

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A096836 Number of primes in the neighborhood of 10^n with radius ceiling(log(10^n)).

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A096837 Number of primes in the neighborhood of n^n with radius ceiling(log(n^n)).

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A096838 Number of primes in the neighborhood of prime(n), the n-th prime in the center with radius ceiling(log(prime(n))).

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A217864 Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).

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A217864 Number of prime numbers between floor(nlog(n)) and (n + 1)log(n + 1).