cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A215238 Prime(A215237).

Original entry on oeis.org

2, 3, 113, 1637, 2971, 44293, 305663, 1133071, 370261, 1357201, 46006769, 268119517, 291057379, 3429782117, 10502593103, 10926444583, 87241770619, 226751019497, 1901687257447
Offset: 0

Views

Author

T. D. Noe, Oct 11 2012

Keywords

Comments

We use offset 0 because A215237 uses that offset.
a(n) is least prime(k) such that there are exactly n primes between prime(k)/2 and prime(k+1)/2. - Peter Munn, Oct 22 2017

Crossrefs

Cf. A000040, A080192, A104380, A215237.

Programs

  • Mathematica
    t = Table[PrimePi[Prime[n + 1]/2] - PrimePi[Prime[n]/2], {n, 100000}]; t2 = Flatten[Table[Position[t, n, 1, 1], {n, 0, 8}]]; Prime[t2]

Formula

a(n) = A000040(A215237(n)).

Extensions

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

A215239 Prime number following prime(A215237).

Original entry on oeis.org

3, 5, 127, 1657, 2999, 44351, 305717, 1133131, 370373, 1357333, 46006967, 268119667, 291057563, 3429782399, 10502593369, 10926444847, 87241771051, 226751019863, 1901687257829
Offset: 0

Views

Author

T. D. Noe, Oct 11 2012

Keywords

Comments

We use offset 0 because A215237 uses that offset.

Crossrefs

Cf. A215237, A215238.

Programs

  • Mathematica
    t = Table[PrimePi[Prime[n + 1]/2] - PrimePi[Prime[n]/2], {n, 100000}]; t2 = Flatten[Table[Position[t, n, 1, 1], {n, 0, 8}]]; NextPrime[Prime[t2]]

Extensions

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

A217564 Number of primes between prime(n)/2 and prime(n+1)/2.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1
Offset: 1

Views

Author

Hans Havermann, Oct 06 2012

Keywords

Comments

Conjecture: this sequence is unbounded, as implied by Dickson's conjecture. - Charles R Greathouse IV, Oct 09 2012
Conjecture: 0 appears infinitely often. - Jon Perry, Oct 10 2012
First differences of A079952. - Peter Munn, Oct 19 2017

Examples

			a(30) = 2 because there are two primes between prime(30)/2 [=113/2] and prime(31)/2 [=127/2]; i.e., the numbers 59 and 61.

Links

Crossrefs

Cf. A079952, A102820.
Cf. A215237 (location of first n).
A164368 lists the prime(n) corresponding to the zero terms.

Programs

  • Mathematica
    q = 2; Table[p = q; q = NextPrime[p]; Length[Position[PrimeQ[Range[p + 1, q - 1, 2]/2], True]], {105}]
Table[PrimePi[Prime[n + 1]/2] - PrimePi[Prime[n]/2], {n, 105}] (* Alonso del Arte, Oct 08 2012 *)

Formula

a(n) = pi(prime(n + 1)/2) - pi(prime(n)/2), where pi is the prime counting function and prime(n) is the n-th prime.
Equivalently, a(n) = A079952(n+1) - A079952(n). - Peter Munn, Oct 19 2017
The average order of a(n) is 1/2, that is, a(1) + a(2) + ... + a(n) ~ n/2. - Charles R Greathouse IV, Oct 09 2012
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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