A217564 -id:A217564 - cp's OEIS frontend

A217658 In A217564, prime(i), where i is the index of the start of the first exactly-n terms between successive zeros.

13, 19, 3, 73, 523, 6581, 10753, 231643, 387679, 43103, 255259, 55457, 28751773, 328934789, 278689963, 784284211, 4440915607, 8340839629, 30651695947

Offset: 1

Hans Havermann, Oct 09 2012

nonn

			The first exactly-4 terms between successive zeros in A217564 are '1,1,1,1' beginning at index 21, so a(4) = prime(21) = 73.
The first exactly-5 terms between successive zeros in A217564 are '2,1,1,1,1' beginning at index 99, so a(5) = prime(99) = 523.

Cf. A217564.

A102820 Number of primes between 2prime(n) and 2prime(n+1), where prime(n) is the n-th prime.

Original entry on oeis.org

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

Offset: 1

Ali A. Tanara (tanara(AT)khayam.ut.ac.ir), Feb 27 2005

Number of primes between successive even semiprimes. [Juri-Stepan Gerasimov, May 01 2010]
From Peter Munn, Jun 01 2023: (Start)
First differences of A020900.
A080192 lists prime(n) corresponding to the zero terms.
A104380(k) is prime(n) corresponding to the first occurrence of k as a term.
If a(n) is nonzero, A059786(n) is the smallest and A059788(n+1) the largest of the a(n) enumerated primes. In the tree of primes described in A290183, these primes label the child nodes of prime(n).
Conjecture: the asymptotic proportions of 0's, 1's, ... , k's, ... are 1/3, 2/9, ... , 2^k/3^(k+1), ... .
(End)

			a(15)=3 because there are 3 primes between the doubles of the 15th and 16th primes, that is between 2*47 and 2*53.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Sequences with related analysis: A020900, A059786, A059788, A080192, A104380, A290183.
Cf. A104272, A080359. [Vladimir Shevelev, Aug 24 2009]
Cf. A100484, A010051.
Sequences with similar definitions: A104289, A217564.

a102820 n = a102820_list !! (n-1)
a102820_list =  map (sum . (map a010051)) $
   zipWith enumFromTo a100484_list (tail a100484_list)
-- Reinhard Zumkeller, Apr 29 2012

Table[PrimePi[2 Prime[n+1]]-PrimePi[2 Prime[n]], {n, 150}] (* Zak Seidov *)
Differences[PrimePi[2 Prime[Range[110]]]] (* Harvey P. Dale, Oct 29 2022 *)

a(n) = primepi(2*prime(n+1)) - primepi(2*prime(n)); \\ Michel Marcus, Sep 22 2017

a(n) = A020900(n+1) - A020900(n). - Peter Munn, Jun 01 2023

More terms from Zak Seidov, Feb 28 2005

A079952 Number of primes less than prime(n)/2.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 13, 14, 15, 15, 15, 16, 16, 16, 18, 18, 19, 19, 21, 21, 21, 22, 23, 23, 24, 24, 24, 24, 25, 25, 27, 29, 30, 30, 30, 30, 30, 30, 31, 32, 32, 32, 33, 34, 34, 34, 36, 36, 36, 37, 38, 39, 40

Offset: 1

Reinhard Zumkeller, Jan 19 2003

nonn

Previous name: Number of primes p such that prime(n) mod 2*p < prime(n).

Same as A055930, except for a(2). [Noticed by R. J. Mathar, Dec 15 2008, proved by Andrey Zabolotskiy, Oct 26 2017]

			n = 6: prime(6) = 13 and 2, 3, 5 are less than 13/2, therefore a(6) = 3.

Cf. A000040, A000720, A055377, A055930, A079951, A079953, A217564.

Table[PrimePi[Prime[n]/2], {n, 75}] (* Michael De Vlieger, Sep 20 2017 *)

a(n) = primepi(prime(n)/2); \\ Michel Marcus, Sep 20 2017

A079950(n, a(n) + 1) = prime(n).
Where defined, that is for n > 2, prime(a(n)) = A055377(prime(n)). - Peter Munn, Sep 18 2017
0 with partial sums of A217564. - David A. Corneth, Oct 26 2017 (found earlier by Peter Munn).

New name from Peter Munn, Sep 18 2017

A215237 Least number k for which primepi(prime(k+1)/2) - primepi(prime(k)/2) = n.

Original entry on oeis.org

1, 2, 30, 259, 429, 4612, 26466, 88110, 31545, 104071, 2775456, 14614604, 15793779, 164082567, 476853784, 495207013, 3613011290, 9032608100, 69827848342

Offset: 0

T. D. Noe, Oct 09 2012

See A215238 and A215239 for prime(a(n)) and the next prime.

Equivalently stated, a(n) is least k such that there are exactly n primes between prime(k)/2 and prime(k+1)/2. - Peter Munn, May 20 2019

			For n = 2, the consecutive primes are 113 and 127; there are two primes between 56.5 and 63.5.  For n = 3, the consecutive primes are 1637 and 1657; there are three primes between 818.5 and 828.5.

Cf. A217564, A215238, A215239.

t = Table[PrimePi[Prime[n+1]/2] - PrimePi[Prime[n]/2], {n, 100000}]; Flatten[Table[Position[t, n, 1, 1], {n, 0, 8}]]

a(14)-a(18) from Donovan Johnson, Oct 13 2012

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

A217658 In A217564, prime(i), where i is the index of the start of the first exactly-n terms between successive zeros.

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A102820 Number of primes between 2*prime(n) and 2*prime(n+1), where prime(n) is the n-th prime.

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A079952 Number of primes less than prime(n)/2.

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A215237 Least number k for which primepi(prime(k+1)/2) - primepi(prime(k)/2) = 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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A102820 Number of primes between 2prime(n) and 2prime(n+1), where prime(n) is the n-th prime.

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