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

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

1,3

Comments

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)

Links

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.

Formula

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

Example

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.

Mathematica

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 *)

Prog

(Haskell)
a102820 n = a102820_list !! (n-1)
a102820_list =  map (sum . (map a010051)) $
   zipWith enumFromTo a100484_list (tail a100484_list)
-- Reinhard Zumkeller, Apr 29 2012
(PARI) a(n) = primepi(2*prime(n+1)) - primepi(2*prime(n)); \\ Michel Marcus, Sep 22 2017

Crossrefs

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.

Sequence in context: A003643 A092788 A058062 * A355717 A024317 A024880

Adjacent sequences: A102817 A102818 A102819 * A102821 A102822 A102823

Keyword

easy,nonn

Author

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

Extensions

More terms from Zak Seidov, Feb 28 2005

Status

approved