

A217564


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


5



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
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OFFSET

1,30


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


LINKS

Hans Havermann, Table of n, a(n) for n = 1..10000


FORMULA

a(n) = pi(prime(n + 1)/2)  pi(prime(n)/2), where pi is the prime counting function and prime(n) is the nth 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


EXAMPLE

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.


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 *)


CROSSREFS

Cf. A079952, A102820.
Cf. A215237 (location of first n).
A164368 lists the prime(n) corresponding to the zero terms.
Sequence in context: A089734 A321375 A307831 * A325200 A266909 A276491
Adjacent sequences: A217561 A217562 A217563 * A217565 A217566 A217567


KEYWORD

nonn


AUTHOR

Hans Havermann, Oct 06 2012


STATUS

approved



